A209613 Expansion of q * phi(-q^2)^2 * psi(q^3) * psi(-q^3)^2 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
1, -1, -3, 1, 4, 3, -6, -1, 9, -4, -12, -3, 14, 6, -12, 1, 16, -9, -18, 4, 18, 12, -24, 3, 21, -14, -27, -6, 28, 12, -30, -1, 36, -16, -24, 9, 38, 18, -42, -4, 40, -18, -42, -12, 36, 24, -48, -3, 43, -21, -48, 14, 52, 27, -48, 6, 54, -28, -60, -12, 62, 30
Offset: 1
Examples
G.f. = q - q^2 - 3*q^3 + q^4 + 4*q^5 + 3*q^6 - 6*q^7 - q^8 + 9*q^9 - 4*q^10 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- 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, 2016.
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^3] QPochhammer[ q^12]^2 / QPochhammer[ q^4]^2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *) a[ n_] := If[ n < 1, 0, DivisorSum[ n, # KroneckerSymbol[ -4, #] KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Jun 09 2015 *) -
PARI
{a(n) = if( n<1, 0, sumdiv( n, d, d * kronecker( -4, d) * kronecker( -3, n/d)))}; -
PARI
{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, (-1)^e, p==3, (-3)^e, f = kronecker( 3, p) ; (-1)^(e * (p%12>6)) * (p^(e+1) - f^(e+1)) / (p - f))))}; -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * (eta(x^2 + A) * eta(x^12 + A) / eta(x^4 + A))^2, n))};
Formula
Expansion of eta(q) * eta(q^3) * (eta(q^2) * eta(q^12) / eta(q^4))^2 in powers of q.
Euler transform of period 12 sequence [-1, -3, -2, -1, -1, -4, -1, -1, -2, -3, -1, -4, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = (-3)^e, a(p^e) = (-1)^(e * (p mod 12 > 6)) * (p^(e+1) - f^(e+1)) / (p - f) if p > 3 where f = Kronecker(3, p).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A113421.
G.f.: Sum_{k>0} k * x^k / (1 + x^k + x^(2*k)) * Kronecker(-4, k).
G.f.: Sum_{k>0} k * x^k / (1 - x^k + x^(2*k)) * A209615(k).
a(2*n) = -a(n) unless n=0. a(3*n) = a(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(18*sqrt(3)) = 0.316567... . - Amiram Eldar, Jan 23 2024
Comments