A055733 - OEIS

%I #20 May 31 2017 16:29:17

%S 1,1,10,381,36628,7120505,2443835736,1351396969615,1127288317316008,

%T 1349611750487720817,2230372438317527996620,4930842713588476723120511,

%U 14211567663513739084746570600,52259895270824126097423028107277,240736564755509319272061470644316416

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

%H Alois P. Heinz, <a href="/A055733/b055733.txt">Table of n, a(n) for n = 0..100</a>

%F 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.

%F a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 1.74218173246413..., c = 0.5728782413434... . - _Vaclav Kotesovec_, Aug 20 2014

%F a(n) = (n!)^3 * [z^n] hypergeom([], [1, 1], z)^n. - _Peter Luschny_, May 31 2017

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(b(n-j, i-1)*binomial(n, j)^2/j!, j=0..n)))

%p     end:

%p a:= n-> n!*b(n$2):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 21 2014

%p 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

%t 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_ *)

%t Table[SeriesCoefficient[HypergeometricPFQ[{}, {1,1}, x]^n, {x,0,n}] n!^3, {n,0,14}] (* _Peter Luschny_, May 31 2017 *)

%Y Cf. A033935.

%Y Column k=3 of A245397.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Jun 09 2000

%E a(0)=1 inserted by _Alois P. Heinz_, Jul 21 2014