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A229326 Total sum of 4th powers of parts in all partitions of n.

%I A229326 #24 May 28 2018 02:50:51
%S A229326 0,1,18,101,392,1119,2904,6407,13578,26218,49218,86782,150860,249723,
%T A229326 408810,647170,1013278,1545029,2337738,3460218,5086658,7350874,
%U A229326 10549872,14929931,21009874,29205500,40385036,55289000,75309056,101692923,136710130,182377824
%N A229326 Total sum of 4th powers of parts in all partitions of n.
%C A229326 The bivariate g.f. for the partition statistic "sum of 4th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^4}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - _Emeric Deutsch_, Dec 06 2015
%C A229326 Convolution of A001159 and A000041. - _Vaclav Kotesovec_, May 28 2018
%H A229326 Alois P. Heinz, <a href="/A229326/b229326.txt">Table of n, a(n) for n = 0..1000</a>
%H A229326 Guo-Niu Han, <a href="https://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO], 2008.
%F A229326 a(n) = Sum_{k=1..n} A066633(n,k) * k^4.
%F A229326 G.f.: g(x) = (Sum_{k>=1} k^4*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229326 a(n) ~ 216*sqrt(2)*Zeta(5)/Pi^5 * exp(Pi*sqrt(2*n/3)) * n^(3/2). - _Vaclav Kotesovec_, May 28 2018
%p A229326 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229326       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229326       ((g, h)-> g+h+[0, h[1]*i^4])(b(n, i-1), b(n-i, i)))))
%p A229326     end:
%p A229326 a:= n-> b(n, n)[2]:
%p A229326 seq(a(n), n=0..40);
%p A229326 # second Maple program:
%p A229326 g := (sum(k^4*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # _Emeric Deutsch_, Dec 06 2015
%t A229326 (* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 0; a[n_] := Sum[T[n, k]*k^4, {k, 1, n}]; Array[a, 32, 0] (* _Jean-François Alcover_, Dec 15 2016 *)
%t A229326 Table[Sum[DivisorSigma[4, k]*PartitionsP[n-k], {k, 1, n}], {n, 0, 40}] (* _Vaclav Kotesovec_, May 27 2018 *)
%Y A229326 Column k=4 of A213191.
%K A229326 nonn
%O A229326 0,3
%A A229326 _Alois P. Heinz_, Sep 20 2013