A132977 Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.

1, 2, 5, 12, 26, 50, 92, 168, 295, 496, 818, 1332, 2126, 3324, 5126, 7824, 11793, 17548, 25857, 37788, 54734, 78578, 111968, 158496, 222842, 311224, 432095, 596676, 819504, 1119624, 1522282, 2060448, 2776514, 3725294, 4978142, 6626988, 8789042

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^(-2/3) * (chi(q) * chi(q^3))^2 * c(q^2) / (3 * b(q^2)) in powers of q where chi() is a Ramanujan theta function and b(), c() are cubic AGM functions.

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

Expansion of (chi^3(q^3) / chi(q))^2 * (psi(-q^3) / psi(-q))^4 in powers of q where chi(), psi() are Ramanujan theta functions.

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

G.f. = A112173(x) * A128758(x^2).

G.f.: (Product_{k>0} (1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2.

a(n) = A132975(3*n + 1).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 50*x^5 + 92*x^6 + 168*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 12*q^10 + 26*q^13 + 50*q^16 + 92*q^19 + ...

Mathematica

nmax = 40; CoefficientList[Series[Product[((1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x^6]^4 / (QPochhammer[ x] QPochhammer[ x^3] QPochhammer[ x^4] QPochhammer[ x^12]))^2, {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

Prog

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

Crossrefs

Cf. A112173, A128758, A132975.

Sequence in context: A338792 A221720 A258099 * A027927 A221948 A116717

Adjacent sequences: A132974 A132975 A132976 * A132978 A132979 A132980

Keyword

nonn

Author

Michael Somos, Sep 07 2007

Extensions

Edited by R. J. Mathar and N. J. A. Sloane, Sep 01 2009

Status

approved