OFFSET
0,3
COMMENTS
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))).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.
FORMULA
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.
MAPLE
b:= proc(n) option remember; add(
(-1)^(n/d+1)*d^6, d=numtheory[divisors](n))
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(b(k)*a(n-k), k=1..n)/n)
end:
seq(a(n), n=0..35); # Alois P. Heinz, Oct 16 2017
MATHEMATICA
nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5), {k, 1, nmax}], {x, 0, nmax}], x]
PROG
(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ G. C. Greubel, Oct 31 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 3012018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 05 2015
STATUS
approved