A248884 - OEIS

%I #32 Sep 24 2022 09:30:35

%S 1,1,32,275,1763,12421,85808,561074,3535678,21815897,131733641,

%T 778099521,4505634324,25635135074,143507764032,791243636305,

%U 4300983535471,23070300486656,122213931799869,639848848696540,3312824859756453,16972058378914997,86082216143323410

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

%C 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 Vaclav Kotesovec, <a href="/A248884/b248884.txt">Table of n, a(n) for n = 0..1000</a>

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

%p       (-1)^(n/d+1)*d^6, 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^5),{k,1,nmax}],{x,0,nmax}],x]

%o (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 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // _G. C. Greubel_, Oct 3012018

%Y Cf. A026007 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).

%Y Column k=5 of A284992.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 05 2015