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
Keywords
Examples
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 + ...
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
Programs
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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 *) -
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))};
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.
Comments