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%I #9 Nov 09 2017 03:34:17
%S 1,1,5,19,54,165,467,1317,3599,9687,25519,66203,169254,426750,1062950,
%T 2616818,6373911,15369774,36716706,86939235,204152395,475631501,
%U 1099874363,2525418842,5759549109,13050991205,29391523405,65801951770,146486952644,324340095729,714389015139
%N Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).
%C Weigh transform of square pyramidal numbers (A000330).
%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/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F G.f.: Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).
%F a(n) ~ exp(5*(15*Zeta(5))^(1/5) * n^(4/5) / 2^(11/5) + 7*Pi^4 * n^(3/5) / (360*2^(2/5) * (15*Zeta(5))^(3/5)) + (Zeta(3) / (2^(13/5) * (15*Zeta(5))^(2/5)) - 49*Pi^8 / (2160000 * 2^(3/5) * 15^(2/5) * Zeta(5)^(7/5)))*n^(2/5) + (343*Pi^12 / (9720000000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + 49*Pi^8 * Zeta(3) / (129600000 * Zeta(5)^2) - 2401 * Pi^16 / (83980800000000 * Zeta(5)^3) - Zeta(3)^2 / (1200*Zeta(5))) * (3*Zeta(5))^(1/10) / (2^(11/18) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - _Vaclav Kotesovec_, Nov 09 2017
%t nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000330, A027998, A258343.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 16 2017
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