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%I #14 Nov 14 2017 17:41:30
%S 1,1,8,29,89,301,915,2763,8040,22910,63776,174174,467448,1233836,
%T 3209679,8234149,20857621,52206847,129227514,316543962,767767628,
%U 1844925743,4394337797,10379319118,24320964976,56557678603,130571770387,299357973400,681777058604,1542840256421,3470045577372
%N Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-2)).
%C Weigh transform of the octagonal numbers (A000567).
%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-2), g(n) = -1. - _Seiichi Manyama_, Nov 14 2017
%H Seiichi Manyama, <a href="/A294838/b294838.txt">Table of n, a(n) for n = 0..8270</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%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>
%F G.f.: Product_{k>=1} (1 + x^k)^A000567(k).
%F a(n) ~ exp(-1800*Zeta(3)^3 / (49*Pi^8) - (9 * 2^(3/4) * 5^(5/4) * Zeta(3)^2 / (7^(5/4)*Pi^5)) * n^(1/4) - (3*sqrt(10/7) * Zeta(3) / Pi^2) * sqrt(n) + (2*(14/5)^(1/4) * Pi/3) * n^(3/4)) * 7^(1/8) / (2^(41/24) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 10 2017
%F a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-2)*(-1)^(1+n/d). - _Seiichi Manyama_, Nov 14 2017
%t nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (3 d - 2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]
%Y Cf. A000567, A027998, A028377, A294102, A294836, A294837.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 09 2017
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