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
Keywords
Programs
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Mathematica
Table[n! * Sum[Binomial[n-1,n-4*k]/k!, {k,0,n/4}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *) With[{nn=20},CoefficientList[Series[Exp[(x/(1-x))^4],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 01 2025 *) -
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;
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