%I #7 Nov 10 2017 11:04:41
%S 1,1,2,7,13,30,61,125,250,494,960,1835,3487,6520,12105,22239,40515,
%T 73207,131315,233831,413625,727100,1270405,2207243,3814155,6557164,
%U 11217391,19099932,32375026,54640509,91836697,153739008,256379360,425964293,705197513,1163452547,1913096832,3135609791
%N Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-1)/2)*(1 + x^(2*k))^(k*(3*k+1)/2).
%C Weigh transform of the generalized pentagonal numbers (A001318).
%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/PentagonalNumber.html">Pentagonal Number</a>
%F G.f.: Product_{k>=1} (1 + x^k)^A001318(k).
%F a(n) ~ exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / (3*5^(1/4)) + 9*Zeta(3) * sqrt(5*n/7) / (4*Pi^2) + (7*Pi^6 - 2430*Zeta(3)^2) * (5/7)^(1/4) * n^(1/4) / (336 * sqrt(2) * Pi^5) + 15*Zeta(3)*(3240*Zeta(3)^2 - 7*Pi^6) / (3136*Pi^8)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 10 2017
%t nmax = 37; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 1)/2) (1 + x^(2 k))^(k (3 k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 37}]
%Y Cf. A001318, A028377, A294102, A294840, A294841.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 09 2017
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