OFFSET
0,2
COMMENTS
In general, for fixed m > 1, if g.f. = Product_{k>=1} (1 - x^(m*k))/(1 - x^k)^m, then a(n, m) ~ exp(Pi*sqrt(2*n*(m-1/m)/3)) * (m^2 - 1)^(m/4) / (2^(3*m/4 + 1/2) * 3^(m/4) * m^(m/4 + 1/2) * n^(m/4 + 1/2)). - Vaclav Kotesovec, Nov 10 2016
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
FORMULA
G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5.
a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(7/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^5; x^5)_inf/((x; x)_inf)^5, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
EXAMPLE
G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 505*x^5 + 1260*x^6 + ...
MAPLE
N:= 100: # to get a(0)..a(N)
S:= series(mul((1-x^(5*n))/(1-x^n)^5, n=1..N), x, N+1):
seq(coeff(S, x, n), n=0..N); # Robert Israel, Nov 09 2016
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^5, x^5]/QPochhammer[x, x]^5 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
PROG
(PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(5*k))/(1-x^k)^5, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016
(PARI) x='x+O('x^66); Vec(eta(x^5)/eta(x)^5) \\ Joerg Arndt, Nov 27 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 07 2016
STATUS
approved