A389289 a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(k,n-2*k).

1, 0, 2, 3, 10, 30, 77, 252, 690, 2145, 6292, 19019, 57421, 173264, 527340, 1600788, 4885698, 14911941, 45623237, 139763525, 428688260, 1316652480, 4047752865, 12457365570, 38371634925, 118295112780, 364968310176, 1126815467736, 3481286021356, 10761998834320

Offset

0,3

Links

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

Formula

a(n) = [x^n] 1/(1 - x^2 - x^3)^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 - x^3) ).

Mathematica

Table[Sum[Binomial[n+k-1, k]Binomial[k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n+k-1, k)*binomial(k, n-2*k));
(Magma) [&+[Binomial(n+k-1, k) * Binomial(k, n-2*k) : k in [0..Floor(n/2)] ]: n in [0..40] ]; // Vincenzo Librandi, Oct 06 2025

Crossrefs

Cf. A217358.

Sequence in context: A319189 A301971 A319671 * A338593 A270363 A131764

Adjacent sequences: A389286 A389287 A389288 * A389290 A389291 A389292

Keyword

nonn

Author

Seiichi Manyama, Sep 28 2025

Status

approved