A229331 - OEIS

%I #23 May 28 2018 02:55:50

%S 0,1,514,20199,283370,2256695,12637956,55247745,202345886,644749920,

%T 1846772550,4836548836,11795957334,27022021703,58819382790,

%U 122237638440,244429962966,471615005229,882955864560,1606698758560,2853601781340,4952029001892,8423307325854

%N Total sum of 9th powers of parts in all partitions of n.

%C The bivariate g.f. for the partition statistic "sum of 9th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^9}*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 Alois P. Heinz, <a href="/A229331/b229331.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 a(n) = Sum_{k=1..n} A066633(n,k) * k^9.

%F G.f.: g(x) = (Sum_{k>=1} k^9*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015

%F a(n) ~ 27648*sqrt(3)/11 * exp(Pi*sqrt(2*n/3)) * n^4. - _Vaclav Kotesovec_, May 28 2018

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),

%p       ((g, h)-> g+h+[0, h[1]*i^9])(b(n, i-1), b(n-i, i)))))

%p     end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=0..40);

%p # second Maple program:

%p g := (sum(k^9*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 (* 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^9, {k, 1, n}]; Array[a, 40, 0] (* _Jean-François Alcover_, Dec 15 2016 *)

%Y Column k=9 of A213191.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 20 2013