A213056 Expansion of chi(x) * f(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.
1, 1, 0, 4, 4, 1, 4, 4, 5, 0, 0, 8, 4, 4, 4, 8, 9, 4, 0, 4, 12, 1, 4, 8, 8, 4, 0, 8, 8, 4, 8, 16, 8, 5, 0, 12, 12, 0, 8, 12, 13, 0, 0, 8, 8, 8, 12, 8, 16, 4, 0, 16, 12, 4, 4, 20, 13, 4, 0, 16, 20, 8, 8, 8, 8, 9, 0, 12, 16, 4, 12, 12, 16, 0, 0, 16, 20, 4, 8
Offset: 0
Keywords
Examples
G.f. = 1 + x + 4*x^3 + 4*x^4 + x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 8*x^11 + 4*x^12 + ... G.f. = q + q^4 + 4*q^10 + 4*q^13 + q^16 + 4*q^19 + 4*q^22 + 5*q^25 + 8*q^34 + ...
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
CoefficientList[QPochhammer[q^2]^2*QPochhammer[-q^3]^3 / (QPochhammer[q] * QPochhammer[q^4]) + O[q]^80, q] (* Jean-François Alcover, Nov 05 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)* eta[q^2]^2*eta[q^6]^9/(eta[q]*eta[q^3]^3*eta[q^4]*eta[q^12]^3), {q, 0, 50}], q] (* G. C. Greubel, Aug 12 2018 *)
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PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^9 / (eta(x + A) * eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A)^3) ,n))}
Formula
Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^6)^9 / (eta(q) * eta(q^3)^3 * eta(q^4) * eta(q^12)^3) in powers of q.
Expansion of q^(-1/9) times theta series of cubic lattice with respect to point [0, 0, 1/3] in powers of q^(1/3).
Euler transform of period 12 sequence [ 1, -1, 4, 0, 1, -7, 1, 0, 4, -1, 1, -3, ...].
G.f.: Product_{k>0} (1 - (-x)^(3*k))^3 * (1 + x^(2*k-1)).
a(4*n + 1) = a(n). a(8*n + 2) = 0.
Comments