A361576 Expansion of e.g.f. exp( (x / (1-x))^4 ).

1, 0, 0, 0, 24, 480, 7200, 100800, 1431360, 21772800, 370137600, 7185024000, 158150361600, 3848298854400, 100865282918400, 2799294930432000, 81599752346112000, 2492894621048832000, 79852538982408192000, 2684220785621286912000

Offset

0,5

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: exp( (x / (1-x))^4 ).

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

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

a(n) = 5*(n-1)*a(n-1) - 10*(n-2)*(n-1)*a(n-2) + 10*(n-3)*(n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)*(5*n - 24)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - Vaclav Kotesovec, Mar 17 2023

Mathematica

Table[n! * Sum[Binomial[n-1, n-4*k]/k!, {k, 0, n/4}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^4)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, (-1)^(j-4)*j*binomial(-4, j-4)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Cf. A000262, A052887, A361572.

Sequence in context: A263470 A276816 A200979 * A052715 A052725 A360517

Adjacent sequences: A361573 A361574 A361575 * A361577 A361578 A361579

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2023

Status

approved