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 A229327 Total sum of 5th powers of parts in all partitions of n. 2
 0, 1, 34, 279, 1370, 4775, 14196, 35745, 83486, 177120, 358710, 681316, 1257414, 2212343, 3811590, 6344760, 10381686, 16534989, 25994160, 39973360, 60802860, 90875412, 134507694, 196208405, 283895550, 405646460, 575437476, 807778980, 1126478494, 1556675935 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 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^5. 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 a(n) ~ 16*sqrt(3)/7 * exp(Pi*sqrt(2*n/3)) * n^2. - 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^5])(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^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 MATHEMATICA Table[Total[Flatten[IntegerPartitions[n]]^5], {n, 0, 30}] (* Harvey P. Dale, Jun 24 2014 *) (* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k

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Last modified May 19 20:41 EDT 2019. Contains 323410 sequences. (Running on oeis4.)