OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-2/3) * (chi(q) * chi(q^3))^2 * c(q^2) / (3 * b(q^2)) in powers of q where chi() is a Ramanujan theta function and b(), c() are cubic AGM functions.
Euler transform of period 12 sequence [ 2, 2, 4, 4, 2, -4, 2, 4, 4, 2, 2, 0, ...].
Expansion of (chi^3(q^3) / chi(q))^2 * (psi(-q^3) / psi(-q))^4 in powers of q where chi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.
G.f.: (Product_{k>0} (1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2.
a(n) = A132975(3*n + 1).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 50*x^5 + 92*x^6 + 168*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 12*q^10 + 26*q^13 + 50*q^16 + 92*q^19 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[((1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x^6]^4 / (QPochhammer[ x] QPochhammer[ x^3] QPochhammer[ x^4] QPochhammer[ x^12]))^2, {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)))^2, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 07 2007
EXTENSIONS
Edited by R. J. Mathar and N. J. A. Sloane, Sep 01 2009
STATUS
approved