A389291 a(n) = Sum_{k=0..floor(n/4)} binomial(n+k-1,k) * binomial(k,n-4*k).

1, 0, 0, 0, 4, 5, 0, 0, 36, 90, 55, 0, 364, 1365, 1680, 680, 3876, 19380, 35910, 29260, 51359, 265650, 657800, 807300, 966420, 3681405, 11044215, 18123840, 21993136, 55141064, 176924440, 360341520, 503232708, 938237157, 2805855780, 6632172300, 10859526418

Offset

0,5

Links

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

Formula

a(n) = [x^n] 1/(1 - x^4 - x^5)^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^4 - x^5) ). See A365731.

Mathematica

Table[Sum[Binomial[n+k-1, k]Binomial[k, n-4*k], {k, 0, Floor[n/4]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(n+k-1, k)*binomial(k, n-4*k));
(Magma) [&+[Binomial(n+k-1, k) * Binomial(k, n-4*k) : k in [0..Floor(n/4)] ]: n in [0..40] ]; // Vincenzo Librandi, Oct 06 2025

Crossrefs

Cf. A365731.

Sequence in context: A143574 A262400 A387200 * A075424 A200619 A199621

Adjacent sequences: A389288 A389289 A389290 * A389292 A389293 A389294

Keyword

nonn

Author

Seiichi Manyama, Sep 28 2025

Status

approved