A361577 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).

1, 0, 0, 0, 1, 20, 300, 4200, 58835, 849240, 12814200, 203742000, 3430355775, 61363001700, 1168815948300, 23734579869000, 513878948207625, 11850279026586000, 290440507342986000, 7543064638441332000, 206860683821114948625, 5968372055889205462500

Offset: 0

Author

Seiichi Manyama, Mar 16 2023

Keywords

• nonn

Crossrefs

Cf. A000262, A335344, A361573.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)^4))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, (-1)^(j-4)*j*binomial(-4, j-4)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k)/(24^k * k!).

a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} (-1)^(k-4) * k * binomial(-4,k-4) * a(n-k)/(n-k)!.