cp's OEIS Frontend

A294845 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k-1)/2).

%I A294845 #7 Nov 10 2017 10:23:55
%S A294845 1,1,9,39,136,511,1785,6139,20404,66406,211418,660752,2030172,6139231,
%T A294845 18300573,53823451,156344596,448886205,1274840165,3583595734,
%U A294845 9976530997,27520998775,75262394273,204130567402,549318633095,1467178746342,3890697051314,10246833932820,26809705578787,69702402930045
%N A294845 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k-1)/2).
%C A294845 Weigh transform of the octagonal pyramidal numbers (A002414).
%H A294845 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 A294845 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 A294845 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A294845 G.f.: Product_{k>=1} (1 + x^k)^A002414(k).
%F A294845 a(n) ~ exp(-2401 * Pi^16 / (2267481600000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (388800000 * Zeta(5)^2) - Zeta(3)^2 / (400*Zeta(5)) + (343*Pi^12 / (87480000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (6480000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5)*Zeta(3) / (2^(13/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (1080 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(11/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(11/20) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - _Vaclav Kotesovec_, Nov 10 2017
%t A294845 nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t A294845 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1) (2 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]
%Y A294845 Cf. A002414, A258343, A281156, A294838, A294841, A294842, A294843, A294844.
%K A294845 nonn
%O A294845 0,3
%A A294845 _Ilya Gutkovskiy_, Nov 09 2017