A387345 Coefficient of x^n in the expansion of ( (1-x+x^4) / (1-x) )^n.

1, 0, 0, 0, 4, 5, 6, 7, 36, 81, 145, 231, 562, 1339, 2835, 5405, 11316, 25177, 55149, 114475, 238869, 513016, 1114597, 2382524, 5052066, 10786905, 23242531, 50010507, 107127199, 229474680, 493391791, 1062577948, 2286124916, 4915013169, 10577945445, 22794425827

Offset

0,5

Links

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

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n,k) * binomial(n-3*k-1,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) / (1-x+x^4) ). See A215341.

Mathematica

Table[SeriesCoefficient[((1-x+x^4)/(1-x))^n, {x, 0, n}], {n, 0, 35}] (* Vincenzo Librandi, Oct 21 2025 *)

Prog

(PARI) a(n, s=4, t=1, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial((u-t+1)*n-(s-1)*k-1, n-s*k));
(Magma) [&+[Binomial(n, k)*Binomial(n-3*k-1, n-4*k): k in [0..Floor(n/4)]]: n in [0..35]]; // Vincenzo Librandi, Oct 21 2025

Crossrefs

Cf. A088218, A246437, A372413, A387546.

Cf. A215341, A387200.

Sequence in context: A010754 A187807 A051036 * A063673 A105737 A335186

Adjacent sequences: A387342 A387343 A387344 * A387346 A387347 A387348

Keyword

nonn

Author

Seiichi Manyama, Sep 29 2025

Status

approved