A386835 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k-1,n-k).
1, 5, 30, 198, 1375, 9843, 71876, 532220, 3981645, 30023265, 227803642, 1737227682, 13303481035, 102234258623, 787997000640, 6089345072056, 47161769198809, 365986358229645, 2845097133606422, 22151577531840830, 172710278146819959, 1348274852150114251
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Binomial[2*n + 2, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k-1, n-k));
Formula
a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^3 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
a(n) ~ 2^(3*n+5) / (27*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 06 2025