A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
1, 5, 8, 4, 4, 13, 12, 4, 5, 16, 24, 8, 4, 20, 12, 8, 9, 20, 32, 4, 12, 29, 12, 8, 8, 36, 40, 8, 8, 20, 24, 16, 8, 25, 40, 12, 12, 32, 24, 12, 13, 48, 40, 8, 8, 40, 36, 8, 16, 20, 56, 16, 12, 52, 12, 20, 13, 36, 56, 16, 20, 40, 24, 8, 8, 45, 72, 12, 16, 52
Offset: 0
Keywords
Examples
G.f. = 1 + 5*x + 8*x^2 + 4*x^3 + 4*x^4 + 13*x^5 + 12*x^6 + 4*x^7 + 5*x^8 + ... G.f. = q + 5*q^4 + 8*q^7 + 4*q^10 + 4*q^13 + 13*q^16 + 12*q^19 + 4*q^22 + ...
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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Magma
A := Basis( ModularForms( Gamma0(36), 3/2), 210); A[2] + 5*A[5];
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}]; (* Michael Somos, Jan 08 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};
Formula
Expansion of q^(-1/3) * eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^5 * eta(q^4)^5 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [5, -7, 4, -2, 5, -7, 5, -2, 4, -7, 5, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 72^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246927.
2 * a(n) = A246928(3*n + 1).
Comments