A387200 Coefficient of x^n in the expansion of (1+x^4+x^5)^n.

1, 0, 0, 0, 4, 5, 0, 0, 28, 72, 45, 0, 220, 858, 1092, 455, 1820, 9520, 18360, 15504, 20349, 101745, 263340, 336490, 347116, 1115730, 3453450, 5920200, 6835140, 13775580, 43345575, 92035125, 128323260, 200786256, 546031024, 1324730440, 2212367080, 3281681700

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

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 A365730.

Mathematica

Table[SeriesCoefficient[Series[(1+t^4+t^5)^n, {t, 0, n}], n], {n, 0, 40}] (* Vincenzo Librandi, Sep 20 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(n, k)*binomial(k, n-4*k));
(Magma) R<t> := PolynomialRing(Integers()); seq := [ MonomialCoefficient((1 + t^4 + t^5)^n, t^n) : n in [0..40] ]; seq; // Vincenzo Librandi, Sep 20 2025

Crossrefs

Cf. A002426, A176806, A192442.

Cf. A365730, A387125.

Sequence in context: A254294 A143574 A262400 * A389291 A075424 A200619

Adjacent sequences: A387197 A387198 A387199 * A387201 A387202 A387203

Keyword

nonn

Author

Seiichi Manyama, Sep 19 2025

Status

approved