A378566 -id:A378566 - cp's OEIS frontend

A378567 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+3*k-1,n-k).

1, 1, 11, 88, 715, 5951, 50288, 429696, 3702987, 32125390, 280211701, 2454992618, 21588647392, 190444368401, 1684556756320, 14935618142768, 132695019071499, 1181070210132582, 10529299131757754, 94005323670592130, 840373149466892965, 7521508912742542806

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Cf. A002002, A378565, A378566.

Cf. A055991, A367234.

a[n_]:=SeriesCoefficient[ 1/(1 - x/(1 - x)^4)^n,{x,0,n}]; Array[a,22,0] (* Stefano Spezia, Dec 01 2024 *)

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(n+3*k-1, n-k));

a(n) = [x^n] 1/(1 - x/(1 - x)^4)^n.

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

Original entry on oeis.org

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

Seiichi Manyama, Dec 01 2024

nonn

Cf. A002002, A378566, A378567.

Cf. A011270, A362087.

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 *)

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

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

cp's OEIS Frontend

A378567 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+3*k-1,n-k).

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