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URL: https://oeis.org/A263526

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A263526
Expansion of f(x, x)^2 / (f(x^3, x^3) * f(x, x^5)) in powers of x where f(, ) is Ramanujan's general theta function.
2
1, 3, 1, -3, -1, 0, 1, 6, 0, -6, -3, -3, 4, 12, 1, -12, -6, -3, 5, 24, 1, -24, -10, -6, 11, 42, 4, -42, -19, -12, 17, 72, 4, -69, -31, -18, 31, 120, 9, -114, -50, -30, 46, 189, 11, -180, -79, -48, 77, 294, 21, -276, -122, -72, 112, 450, 28, -420, -183, -108
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(x)^3 / (f(-x^2) * f(x^3) * f(-x^6)) in powers of x where f() is a Ramanujan theta function.
Expansion of q^(1/3) * eta(q^2)^8 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 3, -5, 2, -2, 3, -2, 3, -2, 2, -5, 3, 0, ...].
a(n) = (-1)^n * A132301(n). Convolution inverse of A261325.
a(2*n) = A132179(n). a(2*n + 1) = 3 * A092848(n). a(4*n) = A230256(n). a(4*n + 1) = 3 * A233034(n). a(4*n + 2) = A233037(n). a(4*n + 3) = -3 * A216046(n).
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 4 * exp(-Pi/3) * Gamma(3/4)^6 / Gamma(11/12)^3 / Gamma(7/12)^3 = A388941. - Simon Plouffe, Sep 21 2025
EXAMPLE
G.f. = 1 + 3*x + x^2 - 3*x^3 - x^4 + x^6 + 6*x^7 - 6*x^9 - 3*x^10 + ...
G.f. = 1/q + 3*q^2 + q^5 - 3*q^8 - q^11 + q^17 + 6*q^20 - 6*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 / (QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6]), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};
KEYWORD
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AUTHOR
Michael Somos, Oct 19 2015
STATUS
approved