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A229325 Total sum of cubes of parts in all partitions of n. 3
0, 1, 10, 39, 122, 287, 660, 1281, 2486, 4392, 7686, 12628, 20790, 32471, 50694, 76560, 115038, 168333, 245784, 350896, 499620, 699468, 975150, 1341077, 1838550, 2490092, 3361260, 4494084, 5986750, 7909231, 10416300, 13616768, 17745948, 22983345, 29672974 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The bivariate g.f. for the partition statistic "sum of cubes of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^3}*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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..8500

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^3.

G.f.: g(x) = (Sum_{k>=1} k^3*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n. - 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^3])(b(n, i-1), b(n-i, i)))))

    end:

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

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

MATHEMATICA

Table[Total[Flatten[IntegerPartitions[n]^3]], {n, 0, 40}] (* Harvey P. Dale, May 01 2016 *)

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], Function[{g, h}, g + h + {0, h[[1]]*i^3}][b[n, i-1], b[n-i, i]]]]];

a[n_] := b[n, n][[2]];

Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Aug 30 2016, after Alois P. Heinz *)

CROSSREFS

Column k=3 of A213191.

Sequence in context: A188480 A059722 A267748 * A074225 A055514 A055233

Adjacent sequences:  A229322 A229323 A229324 * A229326 A229327 A229328

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2013

STATUS

approved

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Last modified July 29 23:58 EDT 2021. Contains 346346 sequences. (Running on oeis4.)