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

%I A055733 #20 May 31 2017 16:29:17
%S A055733 1,1,10,381,36628,7120505,2443835736,1351396969615,1127288317316008,
%T A055733 1349611750487720817,2230372438317527996620,4930842713588476723120511,
%U A055733 14211567663513739084746570600,52259895270824126097423028107277,240736564755509319272061470644316416
%N A055733 Sum of third powers of coefficients in full expansion of (z1+z2+...+zn)^n.
%H A055733 Alois P. Heinz, <a href="/A055733/b055733.txt">Table of n, a(n) for n = 0..100</a>
%F A055733 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 A055733 a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 1.74218173246413..., c = 0.5728782413434... . - _Vaclav Kotesovec_, Aug 20 2014
%F A055733 a(n) = (n!)^3 * [z^n] hypergeom([], [1, 1], z)^n. - _Peter Luschny_, May 31 2017
%p A055733 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A055733       add(b(n-j, i-1)*binomial(n, j)^2/j!, j=0..n)))
%p A055733     end:
%p A055733 a:= n-> n!*b(n$2):
%p A055733 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 21 2014
%p A055733 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 A055733 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 A055733 Table[SeriesCoefficient[HypergeometricPFQ[{}, {1,1}, x]^n, {x,0,n}] n!^3, {n,0,14}] (* _Peter Luschny_, May 31 2017 *)
%Y A055733 Cf. A033935.
%Y A055733 Column k=3 of A245397.
%K A055733 nonn
%O A055733 0,3
%A A055733 _Vladeta Jovovic_, Jun 09 2000
%E A055733 a(0)=1 inserted by _Alois P. Heinz_, Jul 21 2014