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A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

%I A248882 #45 Jul 31 2025 07:34:36
%S A248882 1,1,8,35,119,433,1476,4962,16128,51367,160105,490219,1476420,4378430,
%T A248882 12805008,36962779,105417214,297265597,829429279,2291305897,
%U A248882 6270497702,17008094490,45744921052,122052000601,323166712109,849453194355,2217289285055,5749149331789
%N A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).
%H A248882 Seiichi Manyama, <a href="/A248882/b248882.txt">Table of n, a(n) for n = 0..5510</a> (terms 0..1000 from Vaclav Kotesovec)
%H A248882 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.
%F A248882 a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
%F A248882 a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 06 2017
%F A248882 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - _Ilya Gutkovskiy_, May 30 2018
%F A248882 Euler transform of A309335. - _Georg Fischer_, Nov 10 2020
%p A248882 b:= proc(n) option remember; add(
%p A248882       (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))
%p A248882     end:
%p A248882 a:= proc(n) option remember; `if`(n=0, 1,
%p A248882       add(b(k)*a(n-k), k=1..n)/n)
%p A248882     end:
%p A248882 seq(a(n), n=0..35);  # _Alois P. Heinz_, Oct 16 2017
%t A248882 nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^3),{k,1,nmax}],{x,0,nmax}],x]
%o A248882 (PARI) x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ _Indranil Ghosh_, Apr 06 2017
%o A248882 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A248882 Cf. A023872, A026007, A027998, A248883, A248884, A309335.
%Y A248882 Column k=3 of A284992.
%K A248882 nonn
%O A248882 0,3
%A A248882 _Vaclav Kotesovec_, Mar 05 2015