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URL: https://oeis.org/A263773

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A263773
Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.
3
1, 6, 9, -12, -42, -18, 36, 48, 45, -12, -108, -36, 84, 84, 72, -72, -186, -54, 36, 120, 126, -96, -216, -72, 180, 186, 126, -12, -336, -90, 216, 192, 189, -144, -324, -144, 84, 228, 180, -168, -540, -126, 288, 264, 252, -72, -432, -144, 372, 342, 279, -216
OFFSET
0,2
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
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(q)^6 / f(q^3)^2 in powers of q where f() is a Ramanujan theta function.
Expansion of (eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^3))^2 in powers of q.
Euler transform of period 12 sequence [ 6, -12, 4, -6, 6, -8, 6, -6, 4, -12, 6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134079.
G.f.: Product_{k>0} (1 - (-x)^k)^6 / (1 - (-x)^(3*k))^2.
a(2*n + 1) = 6 * A252651(n). a(3*n + 2) = 9 * A134079(n).
Convolution square of A226535.
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = Pi^(4/3) * 3^(2/3) * Gamma(11/12)^(2/3) / Gamma(2/3)^(2/3) / Gamma(3/4)^(14/3) / (sqrt(2) * (1+3^(1/2)))^(2/3) = A388943. - Simon Plouffe, Sep 21 2025
EXAMPLE
G.f. = 1 + 6*x + 9*x^2 - 12*x^3 - 42*x^4 - 18*x^5 + 36*x^6 + 48*x^7 + 45*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^6 / QPochhammer[ -q^3]^2, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^3))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 27 2015
STATUS
approved