Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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