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A229331 Total sum of 9th powers of parts in all partitions of n. 2
0, 1, 514, 20199, 283370, 2256695, 12637956, 55247745, 202345886, 644749920, 1846772550, 4836548836, 11795957334, 27022021703, 58819382790, 122237638440, 244429962966, 471615005229, 882955864560, 1606698758560, 2853601781340, 4952029001892, 8423307325854 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

FORMULA

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

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

a(n) ~ 27648*sqrt(3)/11 * exp(Pi*sqrt(2*n/3)) * n^4. - Vaclav Kotesovec, May 28 2018

MAPLE

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

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

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

    end:

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

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

# second Maple program:

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

MATHEMATICA

(* 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-Fran├žois Alcover, Dec 15 2016 *)

CROSSREFS

Column k=9 of A213191.

Sequence in context: A213065 A045054 A301547 * A253637 A231415 A271760

Adjacent sequences:  A229328 A229329 A229330 * A229332 A229333 A229334

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2013

STATUS

approved

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Last modified February 29 09:35 EST 2020. Contains 332355 sequences. (Running on oeis4.)