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 ∑ m , n = 0 ∞ D ⁢ ( m , n ) ⁢ x m ⁢ y n = 1 1 - x - y - x ⁢ y superscript subscript 𝑚 𝑛 0 𝐷 𝑚 𝑛 superscript 𝑥 𝑚 superscript 𝑦 𝑛 1 1 𝑥 𝑦 𝑥 𝑦 {\displaystyle{\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}		 

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]
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26.6.E6 ∑ n = 0 ∞ D ⁢ ( n , n ) ⁢ x n = 1 1 - 6 ⁢ x + x 2 superscript subscript 𝑛 0 𝐷 𝑛 𝑛 superscript 𝑥 𝑛 1 1 6 𝑥 superscript 𝑥 2 {\displaystyle{\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}}}		 

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)]]
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26.6.E7 ∑ n = 0 ∞ M ⁢ ( n ) ⁢ x n = 1 - x - 1 - 2 ⁢ x - 3 ⁢ x 2 2 ⁢ x 2 superscript subscript 𝑛 0 𝑀 𝑛 superscript 𝑥 𝑛 1 𝑥 1 2 𝑥 3 superscript 𝑥 2 2 superscript 𝑥 2 {\displaystyle{\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}}		 

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)]
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26.6.E8 ∑ n , k = 1 ∞ N ⁢ ( n , k ) ⁢ x n ⁢ y k = 1 - x - x ⁢ y - ( 1 - x - x ⁢ y ) 2 - 4 ⁢ x 2 ⁢ y 2 ⁢ x superscript subscript 𝑛 𝑘 1 𝑁 𝑛 𝑘 superscript 𝑥 𝑛 superscript 𝑦 𝑘 1 𝑥 𝑥 𝑦 superscript 1 𝑥 𝑥 𝑦 2 4 superscript 𝑥 2 𝑦 2 𝑥 {\displaystyle{\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}		 

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]
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26.6.E9 ∑ n = 0 ∞ r ⁢ ( n ) ⁢ x n = 1 - x - 1 - 6 ⁢ x + x 2 2 ⁢ x superscript subscript 𝑛 0 𝑟 𝑛 superscript 𝑥 𝑛 1 𝑥 1 6 𝑥 superscript 𝑥 2 2 𝑥 {\displaystyle{\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}		 n ≥ 1 𝑛 1 {\displaystyle{\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]
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26.6.E10 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 {\displaystyle{\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)		 m ≥ 1 , n ≥ 1 formulae-sequence 𝑚 1 𝑛 1 {\displaystyle{\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]
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26.6.E11 M ⁢ ( n ) = M ⁢ ( n - 1 ) + ∑ k = 2 n M ⁢ ( k - 2 ) ⁢ M ⁢ ( n - k ) 𝑀 𝑛 𝑀 𝑛 1 superscript subscript 𝑘 2 𝑛 𝑀 𝑘 2 𝑀 𝑛 𝑘 {\displaystyle{\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)		 n ≥ 2 𝑛 2 {\displaystyle{\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]
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