A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346

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 eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.

G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.

Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].

a(n) = 2 * A233693(n) unless n=0. a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.

a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.

a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};

Crossrefs

Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.

Sequence in context: A051466 A320193 A260215 * A080015 A210030 A080054

Adjacent sequences: A261153 A261154 A261155 * A261157 A261158 A261159

Keyword

nonn

Author

Michael Somos, Aug 10 2015

Status

approved