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%I #9 Nov 09 2017 04:09:54
%S 1,1,6,25,78,258,800,2463,7344,21511,61677,173980,483319,1323470,
%T 3577605,9553658,25227727,65918419,170552866,437196640,1110945961,
%U 2799689792,7000246591,17372882671,42809388080,104774554942,254771953179,615667051237,1478934870484,3532347875968
%N Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).
%C Weigh transform of octahedral numbers (A005900).
%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/OctahedralNumber.html">Octahedral Number</a>
%F G.f.: Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).
%F a(n) ~ exp(-Zeta(3)^2 / (600*Zeta(5)) + (Zeta(3) / (4*(15*Zeta(5))^(2/5))) * n^(2/5) + (5*(15*Zeta(5))^(1/5) / 4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (sqrt(Pi) * 2^(47/90) * 5^(2/5) * n^(3/5)). - _Vaclav Kotesovec_, Nov 09 2017
%t nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A005900, A248882, A258343.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 16 2017
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