OFFSET
0,4
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Michael Somos, A Remarkable eta-product Identity
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q.
Euler transform of a period 60 sequence.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * (1 - u)^2 * (1 - v)^2 + 2 * u*v * (u - v)^2 + u*v * (v - u^2) * (u - v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058728.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))).
Convolution inverse of A132968.
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
G.f. = 1 + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^15, x^15] / (QPochhammer[ -x^6, x^6] QPochhammer[ -x^10, x^10]), {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 25 2008
STATUS
approved