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A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

%I A248883 #34 Sep 08 2022 08:46:10
%S A248883 1,1,16,97,457,2297,11113,52049,235334,1039886,4497930,19074006,
%T A248883 79418883,325184763,1311252535,5212704708,20449320159,79231806015,
%U A248883 303428397505,1149325838320,4308477305997,15993198330782,58815616643170,214383601754107,774837953045873
%N A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).
%H A248883 Seiichi Manyama, <a href="/A248883/b248883.txt">Table of n, a(n) for n = 0..3360</a> (terms 0..1000 from Vaclav Kotesovec)
%H A248883 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 A248883 a(n) ~ 31^(1/12) * exp(1/5 * (31/7)^(1/6) * 6^(2/3) * Pi * n^(5/6)) / (2^(7/6) * 3^(2/3) * 7^(1/12) * n^(7/12)).
%F A248883 a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 06 2017
%F A248883 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(k*(1 - x^k)^5)). - _Ilya Gutkovskiy_, May 30 2018
%p A248883 b:= proc(n) option remember; add(
%p A248883       (-1)^(n/d+1)*d^5, d=numtheory[divisors](n))
%p A248883     end:
%p A248883 a:= proc(n) option remember; `if`(n=0, 1,
%p A248883       add(b(k)*a(n-k), k=1..n)/n)
%p A248883     end:
%p A248883 seq(a(n), n=0..35);  # _Alois P. Heinz_, Oct 16 2017
%t A248883 nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^4),{k,1,nmax}],{x,0,nmax}],x]
%o A248883 (PARI) x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^4))) \\ _Indranil Ghosh_, Apr 06 2017
%o A248883 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^4: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A248883 Cf. A026007, A027998, A248882, A248884.
%Y A248883 Column k=4 of A284992.
%K A248883 nonn
%O A248883 0,3
%A A248883 _Vaclav Kotesovec_, Mar 05 2015