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A257536
Expansion of phi(-x^4)^2 * f(-x^1, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.
2
1, -1, 0, 0, -4, 3, 0, 0, 5, 0, 0, 0, -4, -4, 0, 0, 9, -4, 0, 0, -12, 3, 0, 0, 8, 12, 0, 0, -8, -4, 0, 0, 8, -5, 0, 0, -12, 0, 0, 0, 13, 0, 0, 0, -8, -8, 0, 0, 16, -4, 0, 0, -12, 12, 0, 0, 13, 12, 0, 0, -20, -8, 0, 0, 8, -9, 0, 0, -16, 12, 0, 0, 16, 0, 0, 0
OFFSET
0,5
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^(-1/3) * eta(q) * eta(q^4)^4 * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 0, 0, -4, -1, -1, -1, -2, 0, 0, -1, -5, -1, 0, 0, -2, -1, -1, -1, -4, 0, 0, -1, -3, ...].
a(4*n + 2) = a(4*n + 3) = 0. 2 * a(n) = A116597(3*n + 1).
EXAMPLE
G.f. = 1 - x - 4*x^4 + 3*x^5 + 5*x^8 - 4*x^12 - 4*x^13 + 9*x^16 - 4*x^17 + ...
G.f. = q - q^4 - 4*q^13 + 3*q^16 + 5*q^25 - 4*q^37 - 4*q^40 + 9*q^49 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 4, 0, x^4]^2 EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A)^2), n))};
CROSSREFS
Cf. A116597.
Sequence in context: A030120 A058878 A236289 * A019983 A019984 A187765
KEYWORD
sign
AUTHOR
Michael Somos, Apr 28 2015
STATUS
approved