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%I #9 Nov 14 2017 03:18:10
%S 1,1,2,8,23,64,160,397,968,2372,5714,13617,32007,74396,171222,390629,
%T 883922,1984631,4423528,9790146,21524829,47027558,102135967,220565018,
%U 473743833,1012274948,2152271718,4554344649,9593260912,20118418061,42012556671,87375161720,181001416773
%N Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).
%C Weigh transform of A060354.
%H Alois P. Heinz, <a href="/A294959/b294959.txt">Table of n, a(n) for n = 0..2000</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 + x^k)^A060354(k).
%F a(n) ~ exp(-2401 * Pi^16 / (3499200000000 * Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - 2*Zeta(3)^2 / (25*Zeta(5)) + (-343*Pi^12 / (810000000 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (360000 * 2^(1/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + (3/2)^(1/5) * Zeta(3) / (5*Zeta(5))^(2/5)) * n^(2/5) - (7*Pi^4 / (180 * 2^(4/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(12/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(69/80) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - _Vaclav Kotesovec_, Nov 12 2017
%p g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(((i-2)^2+i)*i/2, j)*g(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> g(n$2):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Nov 12 2017
%t nmax = 32; CoefficientList[Series[Product[(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[(-1)^(k/d + 1) 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, A294958.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 12 2017