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A139216
Expansion of q^(-1) * psi(-q) * phi(-q^9) / (psi(-q^3) * psi(q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
4
1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, -3, 0, -4, 0, 0, 0, 4, 0, 5, 0, 0, 0, -7, 0, -8, 0, 0, 0, 12, 0, 14, 0, 0, 0, -17, 0, -20, 0, 0, 0, 24, 0, 28, 0, 0, 0, -36, 0, -40, 0, 0, 0, 52, 0, 56, 0, 0, 0, -71, 0, -80, 0, 0, 0, 96, 0, 109, 0, 0, 0, -133
OFFSET
-1,11
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 eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 / (eta(q^2) * eta(q^3) * eta(q^12)^3 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, -2, 0, -1, 1, -1, 0, 0, -1, -1, -2, -1, -1, 0, 0, -1, 1, -1, 0, -2, -1, -1, -1, -1, -1, 0, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139214.
a(n) = -(-1)^n * A139215(n). a(2*n) = 0 unless n=0.
a(3*n + 1) = 0. a(6*n + 3) = - A217786(n). - Michael Somos, Sep 07 2015
EXAMPLE
G.f. = 1/q - 1 - q^3 + 2*q^9 + 2*q^11 - 3*q^15 - 4*q^17 + 4*q^21 + 5*q^23 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 4, 0, q^9] / (EllipticTheta[ 2, Pi/4, q^(3/2)] EllipticTheta[ 2, 0, q^3]), {q, 0, n}] // Simplify; (* Michael Somos, Sep 07 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)^2 * eta(x^9 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^3 * eta(x^18 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 11 2008
STATUS
approved