A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.

1, -2, 1, -4, 6, -2, 12, -16, 5, -28, 36, -12, 60, -76, 24, -120, 150, -46, 228, -280, 86, -416, 504, -152, 732, -878, 262, -1252, 1488, -442, 2088, -2464, 725, -3408, 3996, -1168, 5460, -6364, 1852, -8600, 9972, -2886, 13344, -15400, 4436, -20424, 23472

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A261369.

a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = A261369(n).

Convolution square of A139136.

a(2*n) = A263538(n). a(2*n + 1) = -2 * A263528(n).

Example

G.f. = 1 - 2*x + x^2 - 4*x^3 + 6*x^4 - 2*x^5 + 12*x^6 - 16*x^7 + 5*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/2) q^(-1/4) (EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -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^3 + A) * eta(x^4 + A) * eta(x^12 + A) / ( eta(x^2 + A) * eta(x^6 + A)^3 ))^2, n))};

Crossrefs

Cf. A139136, A261325, A261328, A261369, A263528, A263538.

Sequence in context: A375043 A285491 A257640 * A293387 A306015 A171087

Adjacent sequences: A262927 A262928 A262929 * A262931 A262932 A262933

Keyword

sign,changed

Author

Michael Somos, Oct 04 2015

Status

approved