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A133637
Expansion of q^(-1) * psi(-q) / psi(-q^3)^3 in powers of q where psi() is a Ramanujan theta function.
5
1, -1, 0, 2, -3, 0, 4, -6, 0, 10, -12, 0, 20, -24, 0, 36, -45, 0, 64, -78, 0, 112, -132, 0, 189, -222, 0, 308, -363, 0, 492, -576, 0, 778, -900, 0, 1210, -1392, 0, 1844, -2121, 0, 2776, -3180, 0, 4144, -4716, 0, 6114, -6936, 0, 8914, -10098, 0, 12884, -14550
OFFSET
-1,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (3 * c(q^2)) / (c(q) * c(q^4)) in powers of q where c() is a cubic AGM function.
Expansion of eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 2, -1, -1, 0, -1, -1, 2, 0, -1, 2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (3/4)^(1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132974.
a(3*n + 1) = 0. a(3*n) = - A132974(n).
Convolution inverse of A113427.
a(3*n - 1) = A263993(n). - Michael Somos, Oct 31 2015
EXAMPLE
G.f. = 1/q - 1 + 2*q^2 - 3*q^3 + 4*q^5 - 6*q^6 + 10*q^8 - 12*q^9 + 20*q^11 - ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 18 2007
STATUS
approved