OFFSET
0,7
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) * eta(q^12) * eta(q^15) * eta(q^20) / (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) in powers of q.
Euler transform of period 60 sequence [-1, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -2, 0, -1, 1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 2, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -2, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, ...].
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 A132967.
G.f.: Product_{k>0} (1 + x^(6*k)) * (1 + x^(10*k)) / ( (1 + x^k) * (1 + x^(15*k)) ).
a(n) = - A132967(n) unless n=0.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/15)) / (4 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
EXAMPLE
G.f. = 1 - q - q^3 + q^4 - q^5 + 2*q^6 - 2*q^7 + 2*q^8 - 3*q^9 + 4*q^10 - ...
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k)) * (1 + x^(10*k)) / ( (1 + x^k) * (1 + x^(15*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 02 2007
STATUS
approved