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Expansion of Product_{k>=1} (1+x^k)^(k^3).
22

%I #43 Sep 08 2022 08:46:10

%S 1,1,8,35,119,433,1476,4962,16128,51367,160105,490219,1476420,4378430,

%T 12805008,36962779,105417214,297265597,829429279,2291305897,

%U 6270497702,17008094490,45744921052,122052000601,323166712109,849453194355,2217289285055,5749149331789

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

%H Seiichi Manyama, <a href="/A248882/b248882.txt">Table of n, a(n) for n = 0..5510</a> (terms 0..1000 from Vaclav Kotesovec)

%H 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 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 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 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 Euler transform of A309335. - _Georg Fischer_, Nov 10 2020

%p b:= proc(n) option remember; add(

%p (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p add(b(k)*a(n-k), k=1..n)/n)

%p end:

%p seq(a(n), n=0..35); # _Alois P. Heinz_, Oct 16 2017

%t nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^3),{k,1,nmax}],{x,0,nmax}],x]

%o (PARI) x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ _Indranil Ghosh_, Apr 06 2017

%o (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 Cf. A026007, A027998, A248883, A248884, A309335.

%Y Column k=3 of A284992.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 05 2015