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 f(-x^4)^4 * f(-x^6)^2 / f(x^2, x^4)^2 = f(-x^4)^4 * f(-x^2, -x^10)^2 / f(-x^12)^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/2) * sqrt(b(q) / (3 * c(q))) * b(q^2) * c(q^2) in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [ -2, -4, 0, -4, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A118272.
G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 - x^k + x^(2*k))^2.
-2 * a(n) = A252650(2*n + 1).
EXAMPLE
G.f. = 1 - 2*x - 3*x^2 + 8*x^3 - 2*x^4 - 6*x^5 + 14*x^6 - 12*x^7 - 9*x^8 + ...
G.f. = q - 2*q^3 - 3*q^5 + 8*q^7 - 2*q^9 - 6*q^11 + 14*q^13 - 12*q^15 + ...
MATHEMATICA
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/2)* (eta[q]*eta[q^2]*eta[q^6]/eta[q^3])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 07 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A))^2, n))};
(Magma) A := Basis( ModularForms( Gamma0(36), 2), 115); A[2] - 2*A[4] - 3*A[6] + 8*A[8] - 2*A[10];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 22 2015
STATUS
approved