A389324 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-1,n-2*k).

1, 0, 2, 6, 18, 60, 200, 672, 2282, 7800, 26802, 92510, 320496, 1113840, 3881488, 13557936, 47455050, 166402800, 584441606, 2055656322, 7239795378, 25527885900, 90109342512, 318383719016, 1125959070968, 3985235211360, 14116185887000, 50036662344552

Offset

0,3

Links

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

Formula

a(n) = [x^n] (1 + x^2 / (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^2 / (1 - x)^2) ). See A389245.

Mathematica

Table[Sum[Binomial[n, k]Binomial[n-1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 07 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n-1, n-2*k));
(Magma) [&+[Binomial(n, k) * Binomial(n-1, n-2*k) : k in [0..Floor(n/2)] ]: n in [0..40]]; // Vincenzo Librandi, Oct 07 2025

Crossrefs

Cf. A246437, A389325, A389326.

Cf. A005725, A389245.

Sequence in context: A148461 A358251 A002527 * A005566 A005631 A118677

Adjacent sequences: A389321 A389322 A389323 * A389325 A389326 A389327

Keyword

nonn

Author

Seiichi Manyama, Sep 29 2025

Status

approved