A378565 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+k-1,n-k).

1, 1, 7, 43, 271, 1746, 11425, 75615, 504799, 3392953, 22930282, 155664356, 1060710457, 7250779238, 49700101101, 341474150583, 2351032782783, 16216401440106, 112035931072915, 775163096510445, 5370301986029066, 37249469056575504, 258648802856972348

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = [x^n] 1/(1 - x/(1 - x)^2)^n.

a(n) ~ (525 - 32*210^(2/3)/(157*sqrt(105) - 1575)^(1/3) + 4*(210*(157*sqrt(105) - 1575))^(1/3))^(1/6) * ((36 + (1208682 - 28350*sqrt(105))^(1/3)/3 + (6*(7461 + 175*sqrt(105)))^(1/3))^n / (2^(2/3) * 7^(1/3) * sqrt(Pi*n) * 3^(n + 1/6) * 5^(n + 1/3))). - Vaclav Kotesovec, Dec 01 2024

Mathematica

Table[Sum[Binomial[n+k-1, k] * Binomial[n+k-1, 2*k-1], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Dec 01 2024 *)

Prog

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

Crossrefs

Cf. A002002, A378566, A378567.

Cf. A011270, A362087.

Sequence in context: A015451 A194779 A126502 * A286911 A343351 A277188

Adjacent sequences: A378555 A378560 A378561 * A378566 A378567 A378568

Keyword

nonn,new

Author

Seiichi Manyama, Dec 01 2024

Status

approved