OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..2500
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (b(q^2) * c(q^2))^3 / (b(q)^2 * c(q) * b(q^4) * c(q^4)^2) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (1/q) * chi(q) * chi(q^3) * chi(-q^6)^4 / chi(-q)^4 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^6))^6 / (eta(q)^5 * eta(q^3) * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ 5, -1, 6, 0, 5, -6, 5, 0, 6, -1, 5, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
1/q + 5 + 14*q + 36*q^2 + 85*q^3 + 180*q^4 + 360*q^5 + 684*q^6 + 1246*q^7 + ...
MATHEMATICA
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^6/(QP[q]^5*QP[q^3]*QP[q^4]* QP[q^12]^5) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^6 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)^5), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 06 2011
STATUS
approved