24.15: Difference between revisions

Latest revision as of 12:03, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
24.15.E1	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}} `\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	(2t)/(exp(t)+ 1) = sum(G[n]((t)^(n))/(factorial(n)), n = 1..infinity)	Divide[2t,Exp[t]+ 1] == Sum[Subscript[G, n]Divide[(t)^(n),(n)!], {n, 1, Infinity}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
24.15.E2	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{n} = 2(1-2^{n})\BernoullinumberB{n}} `G_{n} = 2(1-2^{n})\BernoullinumberB{n}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	G[n] = 2(1 - (2)^(n))bernoulli(n)	Subscript[G, n] == 2(1 - (2)^(n))BernoulliB[n]	Failure	Failure	Failed [30 / 30] Result: -.1339745960+.5000000000I Test Values: {G[n] = 1/23^(1/2)+1/2I, n = 1} Result: 1.866025404+.5000000000I Test Values: {G[n] = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.1339745962155613, 0.49999999999999994] Test Values: {Rule[n, 1], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.8660254037844388, 0.49999999999999994] Test Values: {Rule[n, 2], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
24.15.E3	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}} `\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	tan(t) = sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity)	Tan[t] == Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [60 / 60] Result: -14.29465634-.1115650801I Test Values: {t = -3/2, T[n] = 1/23^(1/2)+1/2I} Result: -13.98985487-.1932363871I Test Values: {t = -3/2, T[n] = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [60 / 60] Result: Complex[-14.29465633421075, -0.1115650800742149] Test Values: {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-13.989854867097504, -0.1932363870390304] Test Values: {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
24.15.E4	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}} `T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	T[2n - 1] = (- 1)^(n - 1)((2)^(2n)((2)^(2n)- 1))/(2n)bernoulli(2n)	Subscript[T, 2n - 1] == (- 1)^(n - 1)Divide[(2)^(2n)((2)^(2n)- 1),2n]BernoulliB[2n]	Failure	Failure	Failed [10 / 10] Result: -15.13397460+.5000000000I Test Values: {T[2n-1] = 1/23^(1/2)+1/2I, n = 3} Result: -16.50000000+.8660254040I Test Values: {T[2n-1] = -1/2+1/2I3^(1/2), n = 3} ... skip entries to safe data	Failed [29 / 30] Result: Complex[-0.1339745962155613, 0.49999999999999994] Test Values: {Rule[n, 1], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.1339745962155612, 0.49999999999999994] Test Values: {Rule[n, 2], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
24.15.E5	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle T_{2n} = 0} `T_{2n} = 0`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	T[2*n] = 0	Subscript[T, 2*n] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
24.15.E6	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}} `\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	bernoulli(n) = sum((- 1)^(k)(factorial(k)Stirling2(n, k))/(k + 1), k = 0..n)	BernoulliB[n] == Sum[(- 1)^(k)Divide[(k)!StirlingS2[n, k],k + 1], {k, 0, n}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
24.15.E7	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}} `\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	bernoulli(n) = sum((- 1)^(k)binomial(n + 1,k + 1)Stirling2(n + k, k)/(binomial(n + k,k)), k = 0..n)	BernoulliB[n] == Sum[(- 1)^(k)Binomial[n + 1,k + 1]StirlingS2[n + k, k]/(Binomial[n + k,k]), {k, 0, n}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.15.E8	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}} `\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n) = (factorial(n))/(n + 1)	Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}, GenerateConditions->None] == Divide[(n)!,n + 1]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.15.E11	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}} `\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	sum(binomial(n,2k)((5)/(9))^(k)* bernoulli(2k)u[n - 2k], k = 0..floor((n)/(2))) = (n)/(6)v[n - 1]+(n)/((3)^(n))v[2n - 2]	Sum[Binomial[n,2k](Divide[5,9])^(k)* BernoulliB[2k]Subscript[u, n - 2k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[n,6]Subscript[v, n - 1]+Divide[n,(3)^(n)]Subscript[v, 2n - 2]	Failure	Failure	Failed [300 / 300] Result: .4330127020+.2500000000I Test Values: {u[n-2k] = 1/23^(1/2)+1/2I, v[n-1] = 1/23^(1/2)+1/2I, v[2n-2] = 1/23^(1/2)+1/2I, n = 1} Result: .4650877169+.2685185185I Test Values: {u[n-2k] = 1/23^(1/2)+1/2I, v[n-1] = 1/23^(1/2)+1/2I, v[2n-2] = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.43301270189221935, 0.24999999999999997] Test Values: {Rule[n, 1], Rule[Subscript[u, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-2, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.46508771684719863, 0.2685185185185185] Test Values: {Rule[n, 2], Rule[Subscript[u, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-2, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
24.15.E12	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}} `\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}`	Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }	sum(binomial(n,2k)((5)/(4))^(k)* euler(2k)v[n - 2*k], k = 0..floor((n)/(2))) = (1)/((2)^(n - 1))	Sum[Binomial[n,2k](Divide[5,4])^(k)* EulerE[2k]Subscript[v, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[1,(2)^(n - 1)]	Missing Macro Error	Failure	-	Failed [29 / 30] Result: Complex[-0.1339745962155613, 0.49999999999999994] Test Values: {Rule[n, 1], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.7165063509461097, -0.12499999999999999] Test Values: {Rule[n, 2], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data

@@ Line 14: / Line 14: @@
 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
 |- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/24.15.E1 24.15.E1] || [[Item:Q7558|<math>\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(2*t)/(exp(t)+ 1) = sum(G[n]*((t)^(n))/(factorial(n)), n = 1..infinity)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[2*t,Exp[t]+ 1] == Sum[Subscript[G, n]*Divide[(t)^(n),(n)!], {n, 1, Infinity}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+| [https://dlmf.nist.gov/24.15.E1 24.15.E1] || <math qid="Q7558">\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(2*t)/(exp(t)+ 1) = sum(G[n]*((t)^(n))/(factorial(n)), n = 1..infinity)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[2*t,Exp[t]+ 1] == Sum[Subscript[G, n]*Divide[(t)^(n),(n)!], {n, 1, Infinity}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |-
-| [https://dlmf.nist.gov/24.15.E2 24.15.E2] || [[Item:Q7559|<math>G_{n} = 2(1-2^{n})\BernoullinumberB{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>G_{n} = 2(1-2^{n})\BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>G[n] = 2*(1 - (2)^(n))*bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[G, n] == 2*(1 - (2)^(n))*BernoulliB[n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1339745960+.5000000000*I
+| [https://dlmf.nist.gov/24.15.E2 24.15.E2] || <math qid="Q7559">G_{n} = 2(1-2^{n})\BernoullinumberB{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>G_{n} = 2(1-2^{n})\BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>G[n] = 2*(1 - (2)^(n))*bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[G, n] == 2*(1 - (2)^(n))*BernoulliB[n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1339745960+.5000000000*I
 Test Values: {G[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.866025404+.5000000000*I
 Test Values: {G[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1339745962155613, 0.49999999999999994]
@@ Line 22: / Line 22: @@
 Test Values: {Rule[n, 2], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/24.15.E3 24.15.E3] || [[Item:Q7560|<math>\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>tan(t) = sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Tan[t] == Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -14.29465634-.1115650801*I
+| [https://dlmf.nist.gov/24.15.E3 24.15.E3] || <math qid="Q7560">\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>tan(t) = sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Tan[t] == Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -14.29465634-.1115650801*I
 Test Values: {t = -3/2, T[n] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -13.98985487-.1932363871*I
 Test Values: {t = -3/2, T[n] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-14.29465633421075, -0.1115650800742149]
@@ Line 28: / Line 28: @@
 Test Values: {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/24.15.E4 24.15.E4] || [[Item:Q7561|<math>T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[2*n - 1] = (- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, 2*n - 1] == (- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -15.13397460+.5000000000*I
+| [https://dlmf.nist.gov/24.15.E4 24.15.E4] || <math qid="Q7561">T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[2*n - 1] = (- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, 2*n - 1] == (- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -15.13397460+.5000000000*I
 Test Values: {T[2*n-1] = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -16.50000000+.8660254040*I
 Test Values: {T[2*n-1] = -1/2+1/2*I*3^(1/2), n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1339745962155613, 0.49999999999999994]
@@ Line 34: / Line 34: @@
 Test Values: {Rule[n, 2], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/24.15.E5 24.15.E5] || [[Item:Q7562|<math>T_{2n} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>T_{2n} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">T[2*n] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[T, 2*n] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+| [https://dlmf.nist.gov/24.15.E5 24.15.E5] || <math qid="Q7562">T_{2n} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>T_{2n} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">T[2*n] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[T, 2*n] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |-
-| [https://dlmf.nist.gov/24.15.E6 24.15.E6] || [[Item:Q7563|<math>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n) = sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n] == Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/24.15.E6 24.15.E6] || <math qid="Q7563">\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n) = sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n] == Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/24.15.E7 24.15.E7] || [[Item:Q7564|<math>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n) = sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/(binomial(n + k,k)), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n] == Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/(Binomial[n + k,k]), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/24.15.E7 24.15.E7] || <math qid="Q7564">\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n) = sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/(binomial(n + k,k)), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n] == Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/(Binomial[n + k,k]), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/24.15.E8 24.15.E8] || [[Item:Q7565|<math>\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n) = (factorial(n))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}, GenerateConditions->None] == Divide[(n)!,n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/24.15.E8 24.15.E8] || <math qid="Q7565">\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n) = (factorial(n))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}, GenerateConditions->None] == Divide[(n)!,n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/24.15.E11 24.15.E11] || [[Item:Q7568|<math>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(binomial(n,2*k)*((5)/(9))^(k)* bernoulli(2*k)*u[n - 2*k], k = 0..floor((n)/(2))) = (n)/(6)*v[n - 1]+(n)/((3)^(n))*v[2*n - 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[n,2*k]*(Divide[5,9])^(k)* BernoulliB[2*k]*Subscript[u, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[n,6]*Subscript[v, n - 1]+Divide[n,(3)^(n)]*Subscript[v, 2*n - 2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4330127020+.2500000000*I
+| [https://dlmf.nist.gov/24.15.E11 24.15.E11] || <math qid="Q7568">\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(binomial(n,2*k)*((5)/(9))^(k)* bernoulli(2*k)*u[n - 2*k], k = 0..floor((n)/(2))) = (n)/(6)*v[n - 1]+(n)/((3)^(n))*v[2*n - 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[n,2*k]*(Divide[5,9])^(k)* BernoulliB[2*k]*Subscript[u, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[n,6]*Subscript[v, n - 1]+Divide[n,(3)^(n)]*Subscript[v, 2*n - 2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4330127020+.2500000000*I
 Test Values: {u[n-2*k] = 1/2*3^(1/2)+1/2*I, v[n-1] = 1/2*3^(1/2)+1/2*I, v[2*n-2] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4650877169+.2685185185*I
 Test Values: {u[n-2*k] = 1/2*3^(1/2)+1/2*I, v[n-1] = 1/2*3^(1/2)+1/2*I, v[2*n-2] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.43301270189221935, 0.24999999999999997]
@@ Line 48: / Line 48: @@
 Test Values: {Rule[n, 2], Rule[Subscript[u, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-2, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/24.15.E12 24.15.E12] || [[Item:Q7569|<math>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(binomial(n,2*k)*((5)/(4))^(k)* euler(2*k)*v[n - 2*k], k = 0..floor((n)/(2))) = (1)/((2)^(n - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[n,2*k]*(Divide[5,4])^(k)* EulerE[2*k]*Subscript[v, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[1,(2)^(n - 1)]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1339745962155613, 0.49999999999999994]
+| [https://dlmf.nist.gov/24.15.E12 24.15.E12] || <math qid="Q7569">\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(binomial(n,2*k)*((5)/(4))^(k)* euler(2*k)*v[n - 2*k], k = 0..floor((n)/(2))) = (1)/((2)^(n - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[n,2*k]*(Divide[5,4])^(k)* EulerE[2*k]*Subscript[v, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[1,(2)^(n - 1)]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1339745962155613, 0.49999999999999994]
 Test Values: {Rule[n, 1], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.7165063509461097, -0.12499999999999999]
 Test Values: {Rule[n, 2], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |}
 </div>

24.15: Difference between revisions

Latest revision as of 12:03, 28 June 2021

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