Combinatorial Analysis - 26.6 Other Lattice Path Numbers
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 26.6.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy}}
\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sum(sum((sum(binomial(n,k)*binomial(m + n - k,n), k = 0..n))*(x)^(m)* (y)^(n), n = 0..infinity), m = 0..infinity) = (1)/(1 - x - y - x*y) |
Sum[Sum[(Sum[Binomial[n,k]*Binomial[m + n - k,n], {k, 0, n}, GenerateConditions->None])*(x)^(m)* (y)^(n), {n, 0, Infinity}, GenerateConditions->None], {m, 0, Infinity}, GenerateConditions->None] == Divide[1,1 - x - y - x*y] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}}}
\sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sum(D(n , n)* (x)^(n), n = 0..infinity) = (1)/(sqrt(1 - 6*x + (x)^(2))) |
Sum[D[n , n]* (x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[1 - 6*x + (x)^(2)]] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}}
\sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sum((sum(((- 1)^(k))/(n + 2 - k)*binomial(n,k)*binomial(2*n + 2 - 2*k,n + 1 - k), k = 0..n))*(x)^(n), n = 0..infinity) = (1 - x -sqrt(1 - 2*x - 3*(x)^(2)))/(2*(x)^(2)) |
Sum[(Sum[Divide[(- 1)^(k),n + 2 - k]*Binomial[n,k]*Binomial[2*n + 2 - 2*k,n + 1 - k], {k, 0, n}, GenerateConditions->None])*(x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1 - x -Sqrt[1 - 2*x - 3*(x)^(2)],2*(x)^(2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n,k=1}^{\infty}N(n,k)x^{n}y^{k} = \frac{1-x-xy-\sqrt{(1-x-xy)^{2}-4x^{2}y}}{2x}}
\sum_{n,k=1}^{\infty}N(n,k)x^{n}y^{k} = \frac{1-x-xy-\sqrt{(1-x-xy)^{2}-4x^{2}y}}{2x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sum(sum(((1)/(n)*binomial(n,k)*binomial(n,k - 1))*(x)^(n)* (y)^(k), k = 1..infinity), n = 1..infinity) = (1 - x - x*y -sqrt((1 - x - x*y)^(2)- 4*(x)^(2)* y))/(2*x) |
Sum[Sum[(Divide[1,n]*Binomial[n,k]*Binomial[n,k - 1])*(x)^(n)* (y)^(k), {k, 1, Infinity}, GenerateConditions->None], {n, 1, Infinity}, GenerateConditions->None] == Divide[1 - x - x*y -Sqrt[(1 - x - x*y)^(2)- 4*(x)^(2)* y],2*x] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x}}
\sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | sum((D(n , n)- D(n + 1 , n - 1))*(x)^(n), n = 0..infinity) = (1 - x -sqrt(1 - 6*x + (x)^(2)))/(2*x) |
Sum[(D[n , n]- D[n + 1 , n - 1])*(x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1 - x -Sqrt[1 - 6*x + (x)^(2)],2*x] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1)}
D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m \geq 1, n \geq 1} | (sum(binomial(n,k)*binomial(m + n - k,n), k = 0..n)) = D(m , n - 1)+ D(m - 1 , n)+ D(m - 1 , n - 1) |
(Sum[Binomial[n,k]*Binomial[m + n - k,n], {k, 0, n}, GenerateConditions->None]) == D[m , n - 1]+ D[m - 1 , n]+ D[m - 1 , n - 1] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 26.6.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)}
M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 2} | (sum(((- 1)^(k))/(n + 2 - k)*binomial(n,k)*binomial(2*n + 2 - 2*k,n + 1 - k), k = 0..n)) = M*(n - 1)+ sum(M*(k - 2)*M*(n - k), k = 2..n) |
(Sum[Divide[(- 1)^(k),n + 2 - k]*Binomial[n,k]*Binomial[2*n + 2 - 2*k,n + 1 - k], {k, 0, n}, GenerateConditions->None]) == M*(n - 1)+ Sum[M*(k - 2)*M*(n - k), {k, 2, n}, GenerateConditions->None] |
Skipped - no semantic math | Skipped - no semantic math | - | - |