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%I #43 May 04 2020 14:09:19
%S 1,1,9,30,106,339,1106,3355,10102,29358,83908,234394,644286,1739933,
%T 4631675,12153197,31485413,80576160,203902261,510490213,1265353568,
%U 3106771717,7559844833,18239351931,43650061720,103657177941,244346681972,571930478187,1329655624297,3071230379625,7049750442386,16085170634548,36489192684910
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).
%C Euler transform of the octagonal numbers (A000567).
%H Vaclav Kotesovec, <a href="/A274998/b274998.txt">Table of n, a(n) for n = 0..2000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).
%F a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Nov 08 2017
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p d^2*(3*d-2), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 02 2016
%t nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
%o (Python)
%o from sympy import divisors
%o from sympy.core.cache import cacheit
%o @cacheit
%o def a(n): return 1 if n==0 else sum(sum(d**2*(3*d - 2) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n
%o print([a(n) for n in range(51)]) # _Indranil Ghosh_, Aug 06 2017, after Maple code
%Y Cf. A000294, A000567, A000335, A023871, A278768, A294667, A294691, A294692.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Nov 30 2016
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