A279479 Expansion of f(-x, -x^5) / f(-x^24)^2 in powers of x where f(, ) is Ramanujan's general theta function.
1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 2, -2, 0, 0, 0, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -2, 0, 0, 5, -5, 0, 0, 0, -5, 0, 0, 6, -2, 0, 0, 0, 0, 0, 0, 7, -1, 0, 0, 0, -5, 0, 0, 10, -10, 0, 0, 0, -10, 0, 0, 12
Offset: 0
Keywords
Examples
G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 + 2*x^24 - 2*x^25 - 2*x^29 + ... G.f. = q^-5 - q^-2 - q^10 + q^19 + q^43 - q^58 + 2*q^67 - 2*q^70 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- 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, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6] / QPochhammer[ x^24]^2 , {x, 0, n}]; a[ n_] := SeriesCoefficient[ 2 x^(3/2) QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6] / EllipticTheta[ 2, 0, x^6] , {x, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^24 + A)^2), n))};
Formula
Expansion of chi(-x) * chi(x^3) / psi(x^12) in powers of x where chi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 1, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 3^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279476.
a(4*n) = a(4*n + 1) = a(8*n + 2) = 0. a(8*n) = A096981(n).
Comments