%I #6 Nov 10 2017 10:56:46

%S 1,1,5,13,34,87,212,504,1167,2665,5933,13042,28191,60148,126688,

%T 263821,543414,1108272,2239182,4484482,8907530,17555485,34345465,

%U 66724969,128772908,246951514,470738283,892159198,1681544803,3152656375,5880839454,10916463171,20169007200,37095527149

%N Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-2))*(1 + x^(2*k))^(k*(3*k+2)).

%C Weigh transform of the generalized octagonal numbers (A001082).

%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)^A001082(k+1).

%F a(n) ~ exp(Pi/3 * (7/5)^(1/4) * 2^(3/4) * n^(3/4) + 9*Zeta(3) / (2*Pi^2) * sqrt(5*n/14) - (405*Zeta(3)^2 / (56*Pi^5) + Pi/48) * (10*n/7)^(1/4) + (6075*Zeta(3)^2 / (196*Pi^8) + 15/(224*Pi^2)) * Zeta(3)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 10 2017

%t nmax = 33; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 2)) (1 + x^(2 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 (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

%Y Cf. A001082, A028377, A294838, A294839, A294840.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 09 2017