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

0,2

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..2500

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).

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

Cf. A124815.

Sequence in context: A010430 A191287 A353983 * A283777 A202171 A182531

Adjacent sequences: A260106 A260107 A260108 * A260110 A260111 A260112

Keyword

nonn

Author

Michael Somos, Jul 16 2015

Status

approved