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A263501
Expansion of phi(-x) * f(-x^2)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.
2
1, -2, -3, 7, 0, -3, 7, -12, -6, 12, -2, -3, 12, 0, -9, 13, -12, -9, 12, -12, -6, 13, 0, -6, 24, -12, -6, 24, -14, -15, 12, 0, -9, 12, -24, -9, 19, 0, -12, 24, 0, -12, 36, -24, -9, 19, -12, -12, 24, 0, -9, 12, -36, -15, 24, -14, -9, 36, 0, -18, 24, -12, -18
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 q^(-1/8) * eta(q)^2 * eta(q^2)^2 / eta(q^3) in powers of q.
Euler transform of period 6 sequence [ -2, -4, -1, -4, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 4374^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263527.
-2 * a(n) = A263456(8*n + 1). a(3*n + 2) = -3 * A212907(n). a(9*n + 4) = 0.
EXAMPLE
G.f. = 1 - 2*x - 3*x^2 + 7*x^3 - 3*x^5 + 7*x^6 - 12*x^7 - 6*x^8 + ...
G.f. = q - 2*q^9 - 3*q^17 + 7*q^25 - 3*q^41 + 7*q^49 - 12*q^57 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3 / QPochhammer[ x^3], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^2 / eta(x^3 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 19 2015
STATUS
approved