A258292 Expansion of psi(-q)^2 * chi(q^3)^2 in powers of q where psi(), f() are Ramanujan theta functions.

1, -2, 1, 0, -2, 2, 0, 0, 1, 4, -4, 0, 0, -4, 0, 0, -2, 2, 4, 0, 2, 0, 0, 0, 0, -6, 2, 0, 0, 2, 0, 0, 1, 0, -4, 0, 4, -4, 0, 0, -4, 2, 0, 0, 0, 8, 0, 0, 0, -2, 3, 0, -4, 2, 0, 0, 0, 0, -4, 0, 0, -4, 0, 0, -2, 4, 0, 0, 2, 0, 0, 0, 4, -4, 2, 0, 0, 0, 0, 0, 2, 4

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

Expansion of f(x*w, x/w)^2 in powers of x where w is a primitive cube root of unity and f() is Ramanujan's general theta function.

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122856.

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

a(n) = (-1)^n * A258279(n). Convolution square of A089807.

a(2*n) = A258228(n). a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = A122856(n). a(4*n) = a(n). a(4*n + 3) = 0. a(12*n + 1) = -2 * A002175(n).

a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

Example

G.f. = 1 - 2*q + q^2 - 2*q^4 + 2*q^5 + q^8 + 4*q^9 - 4*q^10 - 4*q^13 + ...kkj

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(3/2)]), {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - 2*A[2] + A[3] - 2*A[5] + 2*A[6] + A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 2*A[18
] + 4*A[19];

Crossrefs

Cf. A002175, A004018, A089807, A122856, A122865, A258228, A258279.

Sequence in context: A134663 A000925 A258279 * A003985 A328948 A287524

Adjacent sequences: A258289 A258290 A258291 * A258293 A258294 A258295

Keyword

sign

Author

Michael Somos, May 25 2015

Status

approved