A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.
1, 2, 1, 0, 0, 2, 3, 2, 2, 0, 0, 2, 3, 2, 0, 0, 0, 0, 2, 4, 1, 0, 0, 2, 2, 2, 4, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 4, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 4, 0, 3, 0, 0, 2, 2, 6, 2, 0, 0, 2, 4, 2, 0, 0, 0, 0, 1, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 4, 0
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + x^2 + 2*x^5 + 3*x^6 + 2*x^7 + 2*x^8 + 2*x^11 + 3*x^12 + ... G.f. = q + 2*q^5 + q^9 + 2*q^21 + 3*q^25 + 2*q^29 + 2*q^33 + 2*q^45 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..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)^(-1/8)*q^(-1/4)*(EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, I*Sqrt[q^3]])^2/(8*EllipticTheta[3, 0, -q^4]*EllipticTheta[2, 0, I*q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^24 + A)), n))};
Formula
Expansion of f(-x^8) * f(x, x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^3)^2 * eta(q^8) * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 0, 0, 2, -1, 2, -1, 0, -2, 2, -2, 2, -2, 0, -1, 2, -1, 2, 0, 0, -2, 2, -2, ...].
Comments