OFFSET
0,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = E(x^5)/E(x)^6 where E(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Dec 05 2010
a(n) ~ 29^(3/2) * exp(sqrt(58*n/15)*Pi) / (2400*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 28 2016
A(x^5) = P(x)*P(a*x)*P(a^2*x)*P(a^3*x)*P(a^4*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n), and where a = exp(2*Pi*i/5) is a primitive fifth root of unity. - Peter Bala, Jan 24 2017
EXAMPLE
G.f.: A(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + ...
log(A(x)) = 6*x + 18*x^2/2 + 24*x^3/3 + 42*x^4/4 + 31*x^5/5 + 72*x^6/6 + 48*x^7/7 + 90*x^8/8 + ... + sigma(5n)*x^n/n + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(5*m)*x^m/m)+x*O(x^n)), n)}
(PARI) default(seriesprecision, 66); Vec(eta(x^5)/eta(x)^6) \\ Joerg Arndt, Dec 05 2010
(PARI) m=30; x='x+O('x^m); Vec(prod(j=1, m+2, (1 - x^(5*j))/(1 - x^j)^6)) \\ G. C. Greubel, Nov 18 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(5*j))/(1 - x^j)^6: j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
s = prod((1 - x^(5*j))/(1 - x^j)^6 for j in (1..32))
list(s) # G. C. Greubel, Nov 18 2018
CROSSREFS
this sequence (k=6).
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Dec 05 2010
STATUS
approved