A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.
1, 2, 2, 4, 5, 6, 6, 4, 7, 4, 6, 8, 4, 10, 8, 12, 8, 6, 14, 8, 11, 6, 8, 8, 8, 14, 6, 12, 15, 14, 14, 8, 12, 14, 12, 16, 8, 10, 14, 16, 16, 12, 12, 12, 16, 10, 10, 8, 19, 20, 20, 8, 12, 24, 14, 24, 12, 16, 14, 16, 21, 10, 14, 28, 16, 12, 14, 12, 16, 16, 30, 12
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 4*x^7 + 7*x^8 + ... G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 6*q^37 + 4*q^43 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- 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[ QPochhammer[ x^2]^2 EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 4, 0, x], {x, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x + A)^2 * eta(x^12 + A)^2), n))};
Formula
Expansion of f(-x^2)^2 * phi(-x^6)^2 / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^6)^4 / (eta(q)^2 * eta(q^12)^2) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263444.
Comments