A385975 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*k,n-k).

1, 1, 5, 16, 61, 231, 896, 3515, 13917, 55501, 222595, 896930, 3628120, 14724022, 59922175, 244456581, 999393021, 4093381925, 16793794625, 69001682216, 283889914171, 1169402930621, 4822302149450, 19905773689763, 82243524966936, 340087268899656, 1407397396315006

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1 + x * (1 + x)^2)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / (1 + x * (1 + x)^2) ). See A161634.

Mathematica

Table[Sum[Binomial[n, k]Binomial[2*k, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*k, n-k));
(Magma) [&+[Binomial(n, k) * Binomial(2*k, n-k) : k in [0..n] ]: n in [0..40]]; // Vincenzo Librandi, Oct 08 2025

Crossrefs

Cf. A002426, A387358, A389225.

Cf. A387288, A389125, A389126.

Cf. A082759, A161634.

Sequence in context: A203414 A189450 A180719 * A343164 A300317 A307469

Adjacent sequences: A385972 A385973 A385974 * A385976 A385977 A385978

Keyword

nonn

Author

Seiichi Manyama, Sep 25 2025

Status

approved