A252651 Expansion of q^(-1/2) * (eta(q) * eta(q^2) * eta(q^6) / eta(q^3))^2 in powers of q.

1, -2, -3, 8, -2, -6, 14, -12, -9, 20, -16, -12, 31, -2, -15, 32, -24, -24, 38, -28, -21, 44, -12, -24, 57, -36, -27, 72, -40, -30, 62, -16, -42, 68, -48, -36, 74, -62, -48, 80, -2, -42, 108, -60, -45, 112, -64, -60, 98, -24, -51, 104, -96, -54, 110, -76, -57

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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 f(-x^4)^4 * f(-x^6)^2 / f(x^2, x^4)^2 = f(-x^4)^4 * f(-x^2, -x^10)^2 / f(-x^12)^2 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-1/2) * sqrt(b(q) / (3 * c(q))) * b(q^2) * c(q^2) in powers of q where b(), c() are cubic AGM theta functions.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A118272.

G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 - x^k + x^(2*k))^2.

-2 * a(n) = A252650(2*n + 1).

Example

G.f. = 1 - 2*x - 3*x^2 + 8*x^3 - 2*x^4 - 6*x^5 + 14*x^6 - 12*x^7 - 9*x^8 + ...
G.f. = q - 2*q^3 - 3*q^5 + 8*q^7 - 2*q^9 - 6*q^11 + 14*q^13 - 12*q^15 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/2)* (eta[q]*eta[q^2]*eta[q^6]/eta[q^3])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 07 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A))^2, n))};
(Magma) A := Basis( ModularForms( Gamma0(36), 2), 115); A[2] - 2*A[4] - 3*A[6] + 8*A[8] - 2*A[10];

Crossrefs

Cf. A118272, A252650.

Sequence in context: A354854 A354861 A348261 * A058485 A204907 A183168

Adjacent sequences: A252648 A252649 A252650 * A252652 A252653 A252654

Keyword

sign

Author

Michael Somos, Mar 22 2015

Status

approved