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A229332 Total sum of 10th powers of parts in all partitions of n.

%I A229332 #23 May 28 2018 02:58:26
%S A229332 0,1,1026,60077,1110704,10936407,72573360,365983991,1513288698,
%T A229332 5365004410,16877063274,48105808222,126584890148,310963328163,
%U A229332 721354362186,1590587613754,3359058693214,6822189191429,13396265918970,25501949210562,47248199227946,85355336473378
%N A229332 Total sum of 10th powers of parts in all partitions of n.
%C A229332 The bivariate g.f. for the partition statistic "sum of 10th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^10}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - _Emeric Deutsch_, Dec 06 2015
%C A229332 In general, column k>0 of A213191 is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - _Vaclav Kotesovec_, May 28 2018
%H A229332 Alois P. Heinz, <a href="/A229332/b229332.txt">Table of n, a(n) for n = 0..1000</a>
%H A229332 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 A229332 a(n) = Sum_{j=k..n} A066633(n,k) * k^10.
%F A229332 G.f.: g(x) = (Sum_{k>=1} k^10*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229332 a(n) ~ 7054387200*sqrt(2)*Zeta(11)/Pi^11 * exp(Pi*sqrt(2*n/3)) * n^(9/2). - _Vaclav Kotesovec_, May 28 2018
%p A229332 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229332       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229332       ((g, h)-> g+h+[0, h[1]*i^10])(b(n, i-1), b(n-i, i)))))
%p A229332     end:
%p A229332 a:= n-> b(n, n)[2]:
%p A229332 seq(a(n), n=0..40);
%p A229332 # second Maple program:
%p A229332 g := (sum(k^10*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 A229332 (* 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^10, {k, 1, n}]; Array[a, 40, 0] (* _Jean-François Alcover_, Dec 15 2016 *)
%Y A229332 Column k=10 of A213191.
%K A229332 nonn
%O A229332 0,3
%A A229332 _Alois P. Heinz_, Sep 20 2013