A260109 Expansion of f(x^3) * f(-x^3)^2 * psi(x)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 3, 4, 6, 9, 12, 14, 12, 16, 18, 18, 24, 21, 27, 28, 30, 36, 24, 38, 42, 40, 42, 36, 48, 43, 48, 52, 48, 54, 60, 62, 54, 56, 66, 72, 72, 74, 63, 72, 78, 81, 84, 64, 84, 88, 84, 90, 72, 98, 108, 100, 102, 72, 108, 110, 114, 112, 96, 126, 96, 133, 120, 104

Offset: 0

Author

Michael Somos, Jul 16 2015

Keywords

• nonn

Comments

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A124815.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 1/4 x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/8) QPochhammer[ -x^3] QPochhammer[ x^3]^2 EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^6 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};

Formula

Expansion of psi(-x^3) * phi(-x^6)^2 * psi(x)^2 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of phi(x) * f(x, x^5) * f(x^2, x^4)^2 in powers of x. - Michael Somos, Jul 18 2015

Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^3) * eta(q^6)^3 / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 3, -2, 2, -1, 3, -6, 3, -1, 2, -2, 3, -4, ...].

a(n) = A124815(2*n + 1). a(3*n + 1) = 3 * a(n).