A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 12, -28, 60, -120, 228, -416, 732, -1252, 2088, -3408, 5460, -8600, 13344, -20424, 30876, -46152, 68268, -100016, 145224, -209120, 298800, -423840, 597108, -835804, 1162824, -1608508, 2212896, -3028632, 4124664, -5590976, 7544604, -10137264, 13565016

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^4 * eta(q^4)^4 * eta(q^6)^10 / ( eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4) in powers of q.

Euler transform of period 12 sequence [ -4, 6, 0, 2, -4, 0, -4, 2, 0, 6, -4, 0, ...].

G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2)^2.

G.f.: (Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)))^2.

a(n) = (-1)^n * A186924(n) = A233673(3*n) = A260215(3*n).

Convolution square of A132002.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 - 4*x + 12*x^2 - 28*x^3 + 60*x^4 - 120*x^5 + 228*x^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10 / (eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};

Crossrefs

Cf. A132002, A186924, A260215, A233673.

Sequence in context: A079089 A182705 A186924 * A179023 A321690 A269712

Adjacent sequences: A261317 A261318 A261319 * A261321 A261322 A261323

Keyword

sign

Author

Michael Somos, Aug 14 2015

Status

approved