A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

1, -1, 0, -2, 1, 0, 2, 0, 0, -2, 2, 0, 1, -1, 0, -2, 0, 0, 2, -2, 0, -2, 0, 0, 3, 0, 0, 0, 2, 0, 2, -2, 0, -2, 0, 0, 2, -1, 0, -2, 1, 0, 0, 0, 0, -4, 2, 0, 2, 0, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 1, 0, 0, -2, 2, 0, 4, 0, 0, -2, 0, 0, 0, -3, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0, 3, -2, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0, 2, 0, 0, -2, 2, 0, 0, 0, 0

Offset

0,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 53 of the 74 eta-quotients listed in Table I of Martin (1996).

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, 2016.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 0, -1, -1, -2, 0, -1, -2, ...].

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e * (e+1) if p == 7 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

a(n) = (-1)^n * A033762(n). a(2*n) = A112604(n). a(2*n + 1) = -A112605(n). a(3*n) = A129451(n). a(3*n + 1) = -a(n). a(3*n + 2) = 0.

a(4*n) = A112606(n). a(4*n + 1) = - A112608(n). a(4*n + 2) = 2 * A112607(n). a(4*n + 3) = - 2 * A112609(n).

a(6*n) = A123884(n). a(6*n + 3) = -2 * A121361(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/12) * exp(Pi / 2) * Pi * 3^(1/2) * Gamma(11/12) * sqrt(2) * (3^(1/2)-1) / Gamma(3/4)^3 / Gamma(2/3) = A388589. - Simon Plouffe, Sep 18 2025

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Dec 13 2025

Example

G.f. = 1 - x - 2*x^3 + x^4 + 2*x^6 - 2*x^9 + 2*x^10 + x^12 - x^13 - 2*x^15 + ...
G.f. = q - q^3 - 2*q^7 + q^9 + 2*q^13 - 2*q^19 + 2*q^21 + q^25 - q^27 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -4, m/d], {d, Divisors[ m]}]]]; (* Michael Somos, Jul 09 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, d) * kronecker( 12, n/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)/ (eta(x^2 + A) * eta(x^6 + A)), n))};

Crossrefs

Cf. A033762, A093766, A112604, A113605, A112606, A112608, A112609, A121361, A123884, A129451, A388589.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A058677 A262780 A033762 * A033798 A033792 A391386

Adjacent sequences: A129446 A129447 A129448 * A129450 A129451 A129452

Keyword

sign,easy

Author

Michael Somos, Apr 16 2007

Status

approved