A121361 Expansion of f(x^1, x^5) * psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 2, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 2, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 0, 2, 2, 1, 3, 0, 0, 0, 1, 0, 0

Offset

0,8

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of q^(-7/12) * eta(q^2) * eta(q^3) * eta(q^4) * eta(q^12) /

(eta(q) * eta(q^6)) in powers of q.

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

2*a(n) = A093829(12*n + 7).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 20 2025

Example

G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + 2*x^7 + x^8 + x^10 + x^11 + ...
G.f. = q^7 + q^19 + q^31 + q^43 + q^67 + q^79 + 2*q^91 + q^103 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Sep 02 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};

Crossrefs

Cf. A093766, A093829.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A127499 A198068 A358194 * A191907 A052343 A073484

Adjacent sequences: A121358 A121359 A121360 * A121362 A121363 A121364

Keyword

nonn

Author

Michael Somos, Jul 16 2006

Status

approved