A139214 Expansion of q * psi(q^2) * psi(-q^9) / (phi(-q^3) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 0, 1, 3, 0, 3, 8, 0, 7, 18, 0, 15, 38, 0, 30, 75, 0, 57, 140, 0, 104, 252, 0, 183, 439, 0, 313, 744, 0, 522, 1232, 0, 852, 1998, 0, 1365, 3182, 0, 2150, 4986, 0, 3336, 7700, 0, 5106, 11736, 0, 7719, 17673, 0, 11538, 26322, 0, 17067, 38808, 0, 25004, 56682, 0

Offset

1,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^4)^2 * eta(q^6)^2 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^3 * eta(q^12) * eta(q^18)) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139216.

a(3*n + 2) = 0. 2 * a(n) = A139213(n) unless n=0.

a(3*n) = A187100(n). a(2*n + 4) = 3 * A261992(n). - Michael Somos, Sep 07 2015

Example

G.f. = q + q^3 + 3*q^4 + 3*q^6 + 8*q^7 + 7*q^9 + 18*q^10 + 15*q^12 + 38*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[(1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 2, Pi/4, q^(9/2)] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A) * eta(x^18 + A)), n))};

Crossrefs

Cf. A139213, A139216, A187100, A261992.

Sequence in context: A127803 A021771 A154853 * A010030 A372339 A197270

Adjacent sequences: A139211 A139212 A139213 * A139215 A139216 A139217

Keyword

nonn

Author

Michael Somos, Apr 11 2008

Status

approved