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A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

%I A248884 #34 Jun 27 2025 23:06:38
%S A248884 1,1,32,275,1763,12421,85808,561074,3535678,21815897,131733641,
%T A248884 778099521,4505634324,25635135074,143507764032,791243636305,
%U A248884 4300983535471,23070300486656,122213931799869,639848848696540,3312824859756453,16972058378914997,86082216143323410
%N A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).
%C A248884 In general, for m > 0, if g.f. = Product_{k>=1} (1+x^k)^(k^m), then a(n) ~ 2^(zeta(-m)) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2))) / (sqrt(2*Pi*(m+2)) * n^((m+3)/(2*m+4))).
%H A248884 Vaclav Kotesovec, <a href="/A248884/b248884.txt">Table of n, a(n) for n = 0..1000</a>
%H A248884 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 A248884 a(n) ~ (5*zeta(7))^(1/14) * 3^(2/7) * exp(zeta(7)^(1/7) * 2^(-9/7) * 3^(-3/7) * 5^(1/7) * 7^(8/7) * n^(6/7)) / (2^(163/252) * 7^(3/7) * sqrt(Pi) * n^(4/7)), where zeta(7) = A013665.
%p A248884 b:= proc(n) option remember; add(
%p A248884       (-1)^(n/d+1)*d^6, d=numtheory[divisors](n))
%p A248884     end:
%p A248884 a:= proc(n) option remember; `if`(n=0, 1,
%p A248884       add(b(k)*a(n-k), k=1..n)/n)
%p A248884     end:
%p A248884 seq(a(n), n=0..35);  # _Alois P. Heinz_, Oct 16 2017
%t A248884 nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5),{k,1,nmax}],{x,0,nmax}],x]
%o A248884 (PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ _G. C. Greubel_, Oct 31 2018
%o A248884 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A248884 Cf. A026007 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).
%Y A248884 Column k=5 of A284992.
%K A248884 nonn
%O A248884 0,3
%A A248884 _Vaclav Kotesovec_, Mar 05 2015