A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.

1, -2, 1, -4, 8, -6, 6, -8, 14, -10, 1, -16, 20, -14, 12, -16, 31, -18, 8, -20, 32, -28, 18, -24, 38, -32, 6, -28, 44, -30, 24, -40, 57, -34, 14, -36, 72, -38, 30, -48, 62, -52, 1, -44, 68, -46, 48, -56, 74, -50, 20, -64, 80, -64, 42, -56, 108, -58, 12, -60, 112, -76, 48, -64, 98, -66, 31, -80, 104, -80, 54, -88

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 f(-x^3)^6 / f(x, x^2)^2 = phi(-x^3)^2 * f(-x, -x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2015

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 16 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252651. - Michael Somos, Mar 22 2015

G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2. - Michael Somos, Mar 22 2015

-3 * a(n) = A118271(3*n + 2).

Example

G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 6*x^6 - 8*x^7 + 14*x^8 + ...
G.f. = q^2 - 2*q^5 + q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 6*q^20 - 8*q^23 + ...

Mathematica

QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[x^3]^6/(QP[-x, x^3]* QP[-x^2, x^3]*QP[x^3])^2, {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 15 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A))^2, n))};
(PARI) q='q+O('q^99); Vec((eta(q)*eta(q^3)*eta(q^6)/eta(q^2))^2) \\ Altug Alkan, Apr 16 2018
(Magma) A := Basis( ModularForms( Gamma0(36), 2), 180); A[3] - 2*A[6] + A[9]; /* Michael Somos, Mar 22 2015 */

Crossrefs

Cf. A118271, A252651.

Sequence in context: A059148 A158451 A257706 * A112173 A058543 A353661

Adjacent sequences: A118269 A118270 A118271 * A118273 A118274 A118275

Keyword

sign

Author

Michael Somos, Apr 21 2006

Status

approved