A055733 Sum of third powers of coefficients in full expansion of (z1+z2+...+zn)^n.

1, 1, 10, 381, 36628, 7120505, 2443835736, 1351396969615, 1127288317316008, 1349611750487720817, 2230372438317527996620, 4930842713588476723120511, 14211567663513739084746570600, 52259895270824126097423028107277, 240736564755509319272061470644316416

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=3 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)^2/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A055733 := proc(n) series(hypergeom([],[1,1],z)^n,z=0,n+1): n!^3*coeff(%,z,n) end: seq(A055733(n), n=0..14); # Peter Luschny, May 31 2017

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^2 / 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 *)
Table[SeriesCoefficient[HypergeometricPFQ[{}, {1,1}, x]^n, {x,0,n}] n!^3, {n,0,14}] (* Peter Luschny, May 31 2017 *)

Formula

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

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 1.74218173246413..., c = 0.5728782413434... . - Vaclav Kotesovec, Aug 20 2014

a(n) = (n!)^3 * [z^n] hypergeom([], [1, 1], z)^n. - Peter Luschny, May 31 2017

Extensions

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