A361576 - OEIS

%I #13 Apr 25 2023 15:23:44

%S 1,0,0,0,24,480,7200,100800,1431360,21772800,370137600,7185024000,

%T 158150361600,3848298854400,100865282918400,2799294930432000,

%U 81599752346112000,2492894621048832000,79852538982408192000,2684220785621286912000

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

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

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

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

%F 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

%t Table[n! * Sum[Binomial[n-1,n-4*k]/k!, {k,0,n/4}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 17 2023 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^4)))

%o (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;

%Y Cf. A000262, A052887, A361572.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Mar 16 2023