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

0,2

Links

Table of n, a(n) for n=0..21.

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

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 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k-1, n-k));

Crossrefs

Cf. A386836, A386837.

Cf. A116881.

Sequence in context: A264910 A196471 A265279 * A034164 A322257 A103433

Adjacent sequences: A386832 A386833 A386834 * A386836 A386837 A386838

Keyword

nonn

Author

Seiichi Manyama, Aug 05 2025

Status

approved