A336578 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(n^2,k-1) for n > 0.

1, 3, 21, 408, 14799, 817743, 61621806, 5921141502, 694008501627, 96176405390961, 15400332946269903, 2799678523675400832, 569877183695866859625, 128436925725088289658534, 31756620986815666396814796, 8548059658831271609064999978, 2488568825786280454788465874035, 779186768737628124697943895022101

Offset: 0

Author

Seiichi Manyama, Jul 26 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..297

Crossrefs

Main diagonal of A336575.

Cf. A336495, A336537, A336577.

Programs

Mathematica

a[0] := 1; a[n_] := Sum[3^k * Binomial[n, k] * Binomial[n^2, k - 1], {k, 1, n}]/n;  Array[a, 18, 0] (* Amiram Eldar, Jul 27 2020 *)

PARI

a(n) = if(n==0, 1, sum(k=1, n, 3^k*binomial(n, k)*binomial(n^2, k-1))/n);

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1)); \\ Seiichi Manyama, Jul 27 2020

PARI

a(n) = sum(k=0, n, 2^k*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1); \\ Seiichi Manyama, Jul 27 2020

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

a(n) = (1/(n^2+1)) * Sum_{k=0..n} 2^k * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).

a(n) ~ 3^n * exp(n - 1/6) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 10 2023

a(n) = 3*hypergeom([1-n, -n^2], [2], 3) for n > 0. - Stefano Spezia, Aug 09 2025