A383481 Coefficient of x^n in the expansion of 1 / (1-x-x^4)^n.

1, 1, 3, 10, 39, 156, 630, 2556, 10431, 42823, 176748, 732810, 3049722, 12732188, 53299284, 223645200, 940355391, 3961092906, 16712516565, 70615352330, 298761296064, 1265504676810, 5366250376710, 22777466596560, 96768003904650, 411451657313931, 1750809473690436, 7455339422353396

Offset: 0

Author

Seiichi Manyama, Apr 28 2025

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1564

Crossrefs

Cf. A003269, A063021, A370624.

Programs

Maple

f:= proc(n) local k; add(binomial(n+k-1,k)*binomial(2*n-3*k-1,n-4*k),k=0..n/4) end proc:
map(f, [$0..40]);  # Robert Israel, May 28 2025

PARI

a(n, s=4, t=1, u=0) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n+k-1,k) * binomial(2*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-x^4) ).