A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.
1, -2, 4, -8, 7, -10, 12, -8, 18, -18, 16, -24, 21, -20, 28, -32, 20, -32, 36, -24, 42, -42, 28, -48, 57, -36, 52, -40, 36, -58, 60, -56, 48, -66, 48, -72, 74, -42, 80, -80, 61, -82, 72, -56, 90, -96, 64, -72, 98, -70, 100, -104, 64, -106, 108, -72, 114, -96
Offset: 0
Keywords
Examples
G.f. = 1 - 2*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ... G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
- Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
- Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
-
Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* Michael Somos, May 19 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))}; -
PARI
{a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};
Formula
Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
Comments