A245669 Expansion of q * f(q, q^5)^3 in powers of q where f() is Ramanujan's two-variable theta function.
1, 3, 3, 1, 0, 3, 6, 3, 3, 6, 6, 3, 0, 6, 6, 1, 6, 9, 6, 0, 0, 12, 12, 3, 7, 6, 9, 6, 0, 12, 6, 3, 6, 6, 12, 3, 0, 12, 12, 6, 6, 12, 18, 6, 0, 12, 12, 3, 7, 15, 12, 0, 0, 9, 12, 6, 12, 18, 6, 6, 0, 18, 18, 1, 12, 12, 18, 6, 0, 12, 12, 9, 12, 18, 15, 6, 0, 18
Offset: 1
Keywords
Examples
G.f. = q + 3*q^2 + 3*q^3 + q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + 6*q^10 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..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[ ((EllipticTheta[ 3, 0, q^(1/3)] - EllipticTheta[ 3, 0, q^3]) / 2)^3, {q, 0, n}]; a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)])^3 / (2^(3/2) q^(1/8)), {q, 0, n}]; -
PARI
{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^3, n))};
Formula
Expansion of q * (chi(q) * psi(-q^3))^3 in powers of q where chi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -3, 0, 0, 3, -3, 3, 0, 0, -3, 3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A245668.
G.f.: x * (Sum_{k in Z} x^(3*k^2 + 2*k))^3.
Convolution cube of A089801.
a(2*n + 2) = A005885(n).
Comments