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

%I A055740 #16 Feb 24 2015 04:11:41
%S A055740 1,1,18,1785,591460,495872505,882463317636,2956241639184631,
%T A055740 17088644286346128840,159584255348964655673745,
%U A055740 2286523844910576580400966980,48220116744252542032928364578451,1446485887751234540636003724054342864,59981372975740557234356339667492583487125
%N A055740 Sum of fourth powers of coefficients in full expansion of (z1+z2+...+zn)^n.
%H A055740 Alois P. Heinz, <a href="/A055740/b055740.txt">Table of n, a(n) for n = 0..100</a>
%F A055740 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.
%F A055740 a(n) ~ c * d^n * (n!)^4 / sqrt(n), where d = 1.511958716403..., c = 0.6632048858... . - _Vaclav Kotesovec_, Aug 20 2014
%p A055740 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A055740       add(b(n-j, i-1)*binomial(n, j)^3/j!, j=0..n)))
%p A055740     end:
%p A055740 a:= n-> n!*b(n$2):
%p A055740 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 21 2014
%t A055740 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_ *)
%Y A055740 Cf. A033935.
%Y A055740 Column k=4 of A245397.
%K A055740 nonn
%O A055740 0,3
%A A055740 _Vladeta Jovovic_, Jun 09 2000
%E A055740 a(0)=1 inserted by _Alois P. Heinz_, Jul 21 2014