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A229327 Total sum of 5th powers of parts in all partitions of n.

%I A229327 #24 May 28 2018 02:52:01
%S A229327 0,1,34,279,1370,4775,14196,35745,83486,177120,358710,681316,1257414,
%T A229327 2212343,3811590,6344760,10381686,16534989,25994160,39973360,60802860,
%U A229327 90875412,134507694,196208405,283895550,405646460,575437476,807778980,1126478494,1556675935
%N A229327 Total sum of 5th powers of parts in all partitions of n.
%C A229327 The bivariate g.f. for the partition statistic "sum of 5th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^5}*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
%H A229327 Alois P. Heinz, <a href="/A229327/b229327.txt">Table of n, a(n) for n = 0..1000</a>
%H A229327 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 A229327 a(n) = Sum_{k=1..n} A066633(n,k) * k^5.
%F A229327 G.f.: g(x) = (Sum_{k>=1} k^5*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229327 a(n) ~ 16*sqrt(3)/7 * exp(Pi*sqrt(2*n/3)) * n^2. - _Vaclav Kotesovec_, May 28 2018
%p A229327 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229327       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229327       ((g, h)-> g+h+[0, h[1]*i^5])(b(n, i-1), b(n-i, i)))))
%p A229327     end:
%p A229327 a:= n-> b(n, n)[2]:
%p A229327 seq(a(n), n=0..40);
%p A229327 # second Maple program:
%p A229327 g := (sum(k^5*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 A229327 Table[Total[Flatten[IntegerPartitions[n]]^5],{n,0,30}] (* _Harvey P. Dale_, Jun 24 2014 *)
%t A229327 (* 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^5, {k, 1, n}]; Array[a, 45, 0] (* _Jean-François Alcover_, Dec 15 2016 *)
%Y A229327 Column k=5 of A213191.
%K A229327 nonn
%O A229327 0,3
%A A229327 _Alois P. Heinz_, Sep 20 2013