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%I #10 Nov 13 2017 04:22:57
%S 1,1,3,9,28,75,198,494,1243,3061,7500,18055,43057,101292,236178,
%T 545218,1248480,2835059,6390360,14298631,31778782,70168935,153993321,
%U 335977369,728962258,1573189113,3377881482,7217395643,15348900996,32494548816,68494383520,143773075158,300568066729
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).
%C Euler transform of A060354.
%H Vaclav Kotesovec, <a href="/A294958/b294958.txt">Table of n, a(n) for n = 0..5000</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 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^A060354(k).
%F a(n) ~ exp(2*Zeta'(-1) + 3*Zeta(3) / (8*Pi^2) - Pi^16 / (1036800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - 2*Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3)/2 + (-Pi^12 / (3600000 * 2^(2/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(2/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(4/5) * 3^(2/5) * Zeta(5)^(7/5)) + 2^(1/5) * Zeta(3) / (3*Zeta(5))^(2/5)) * n^(2/5) - (Pi^4 / (60 * 2^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5*(3*Zeta(5))^(1/5) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(53/400) / (2^(47/200) * sqrt(5*Pi) * n^(253/400)). - _Vaclav Kotesovec_, Nov 12 2017
%p N:=100:
%p S:= series(mul(1/(1 - x^k)^(k*((k-2)^2+k)/2),k=1..N),x,N+1):
%p seq(coeff(S,x,k),k=0..N); # _Robert Israel_, Nov 12 2017
%t nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
%Y Cf. A057145, A060354, A294959.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 12 2017
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