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A210063
Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
6
1, -2, 4, -8, 15, -26, 44, -72, 114, -178, 272, -408, 605, -884, 1276, -1824, 2580, -3616, 5028, -6936, 9498, -12922, 17468, -23472, 31369, -41700, 55156, -72616, 95172, -124202, 161436, -209016, 269616, -346562, 443952, -566856, 721530, -915642, 1158608
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^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5 in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, 2, -2, 3, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A210030.
a(n) = (-1)^n * A187154(n). Convolution inverse of A208589.
a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (16*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
EXAMPLE
1 - 2*x + 4*x^2 - 8*x^3 + 15*x^4 - 26*x^5 + 44*x^6 - 72*x^7 + 114*x^8 + ...
q - 2*q^3 + 4*q^5 - 8*q^7 + 15*q^9 - 26*q^11 + 44*q^13 - 72*q^15 + 114*q^17 + ...
MATHEMATICA
CoefficientList[Series[x^(-1/2) * EllipticTheta[2, 0, x^2] / (2*EllipticTheta[3, 0, x]), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2012
STATUS
approved