A260547 Expansion of psi(x^3) * psi(-x^3) * chi(-x) / phi(-x)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, 3, 8, 19, 41, 83, 160, 296, 530, 923, 1569, 2611, 4264, 6848, 10833, 16904, 26049, 39683, 59817, 89286, 132064, 193683, 281800, 406955, 583577, 831323, 1176841, 1656096, 2317416, 3225472, 4466466, 6154859, 8442088, 11527811, 15674377, 21225403, 28629545

Offset

0,2

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

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-17/24) * eta(q^2) * eta(q^6) * eta(q^12) / eta(q)^3 in powers of q.

Euler transform of period 12 sequence [ 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 0, ...].

2 * a(n) = A001935(3*n + 2).

a(n) ~ exp(Pi*sqrt(3*n/2)) / (2^(15/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 160*x^6 + 296*x^7 + ...
G.f. = q^17 + 3*q^41 + 8*q^65 + 19*q^89 + 41*q^113 + 83*q^137 + 160*q^161 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^6] QPochhammer[ x^12] / QPochhammer[ x]^3, {x, 0, n}];

Prog

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

Crossrefs

Cf. A001935.

Sequence in context: A136396 A006380 A328540 * A328541 A182818 A095846

Adjacent sequences: A260544 A260545 A260546 * A260548 A260549 A260550

Keyword

nonn

Author

Michael Somos, Jul 28 2015

Status

approved