View source for 26.6

{{DISPLAYTITLE:Combinatorial Analysis - 26.6 Other Lattice Path Numbers}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/26.6.E5 26.6.E5] || [[Item:Q7807|<math>\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E6 26.6.E6] || [[Item:Q7808|<math>\sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(D(n , n)* (x)^(n), n = 0..infinity) = (1)/(sqrt(1 - 6*x + (x)^(2)))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[D[n , n]* (x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[1 - 6*x + (x)^(2)]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E7 26.6.E7] || [[Item:Q7809|<math>\sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E8 26.6.E8] || [[Item:Q7810|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\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}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E9 26.6.E9] || [[Item:Q7811|<math>\sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x}</syntaxhighlight> || <math>n \geq 1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((D(n , n)- D(n + 1 , n - 1))*(x)^(n), n = 0..infinity) = (1 - x -sqrt(1 - 6*x + (x)^(2)))/(2*x)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E10 26.6.E10] || [[Item:Q7812|<math>D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1)</syntaxhighlight> || <math>m \geq 1, n \geq 1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(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)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/26.6.E11 26.6.E11] || [[Item:Q7813|<math>M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)</syntaxhighlight> || <math>n \geq 2</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(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)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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