A055740 Sum of fourth powers of coefficients in full expansion of (z1+z2+...+zn)^n.

1, 1, 18, 1785, 591460, 495872505, 882463317636, 2956241639184631, 17088644286346128840, 159584255348964655673745, 2286523844910576580400966980, 48220116744252542032928364578451, 1446485887751234540636003724054342864, 59981372975740557234356339667492583487125

Offset: 0

Author

Vladeta Jovovic, Jun 09 2000

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A033935.

Column k=4 of A245397.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^3/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^3 /j!, {j, 0, n}]]]; a[n_] := n!*b[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

a(n) is coefficient of x^n in expansion of n!^4*(1+x/1!^4+x^2/2!^4+x^3/3!^4+...+x^n/n!^4)^n.

a(n) ~ c * d^n * (n!)^4 / sqrt(n), where d = 1.511958716403..., c = 0.6632048858... . - Vaclav Kotesovec, Aug 20 2014

Extensions

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014