A389329 a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(n+k-1,n-3*k).

1, 0, 0, 3, 16, 50, 135, 413, 1456, 5160, 17490, 58245, 196471, 673244, 2316769, 7955948, 27305840, 93934384, 324040536, 1119701787, 3872489826, 13404527930, 46448444889, 161120715822, 559403953207, 1943722327200, 6758519972100, 23516153724180

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1 + x^3 / (1 - x)^4)^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^3 / (1 - x)^4) ). See A389251.

Mathematica

Table[Sum[Binomial[n, k]Binomial[n+k-1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(n, k)*binomial(n+k-1, n-3*k));
(Magma) [&+[Binomial(n, k) * Binomial(n+k-1, n-3*k) : k in [0..Floor(n/3)] ]: n in [0..40]]; // Vincenzo Librandi, Oct 08 2025

Crossrefs

Cf. A372413, A389327, A389328.

Cf. A389251.

Sequence in context: A346556 A396170 A004320 * A389377 A089363 A000574

Adjacent sequences: A389326 A389327 A389328 * A389330 A389331 A389332

Keyword

nonn

Author

Seiichi Manyama, Sep 29 2025

Status

approved