OFFSET
0,6
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 (chi(q^3) * psi(-q^3)^2)^2 / (psi(-q) * f(q^9)^2) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3 / (eta(q) * eta(q^4) * eta(q^18)^9) in powers of q.
Euler transform of period 36 sequence [ 1, 0, -1, 1, 1, -2, 1, 1, -4, 0, 1, -3, 1, 0, -1, 1, 1, 4, 1, 1, -1, 0, 1, -3, 1, 0, -4, 1, 1, -2, 1, 1, -1, 0, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258111. - Michael Somos, May 20 2015
Convolution inverse of A164616.
a(n) = (-1)^n * A182034(n). - Michael Somos, May 20 2015
EXAMPLE
G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 - 2*q^10 - 4*q^11 - 2*q^13 - 8*q^14 + ...
MATHEMATICA
eta[x_] := QPochhammer[x]; A164615[n_] := SeriesCoefficient[eta[q^2]* eta[q^3]^2*eta[q^9]^3*eta[q^12]^2*eta[q^36]^3/(eta[q]*eta[q^4] *eta[q^18]^9), {q, 0, n}]; Table[A164615[n], {n, 0, 50}] (* G. C. Greubel, Aug 10 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 17 2009
STATUS
approved