A270640 Expansion of psi(x) * psi(x^6) / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 2, 1, 0, 4, 2, 0, 6, 4, 0, 9, 5, 0, 14, 8, 0, 20, 12, 0, 30, 16, 0, 41, 22, 0, 56, 32, 0, 76, 42, 0, 102, 56, 0, 136, 75, 0, 178, 97, 0, 232, 126, 0, 300, 164, 0, 384, 208, 0, 490, 264, 0, 620, 336, 0, 780, 420, 0, 977, 526, 0, 1218, 656, 0, 1512, 810

Offset

0,4

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

Euler transform of period 12 sequence [ 1, -1, 2, -1, 1, 0, 1, -1, 2, -1, 1, -1, ...].

G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 + x^(6*k)).

Example

G.f. = 1 + x + 2*x^3 + x^4 + 4*x^6 + 2*x^7 + 6*x^9 + 4*x^10 + 9*x^12 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 4*q^13 + 2*q^15 + 6*q^19 + 4*q^21 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2^-1 x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] / (QPochhammer[ x^3, x^6] QPochhammer[ x^6, x^12]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];

Prog

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

Crossrefs

Sequence in context: A136329 A122073 A106236 * A139136 A122792 A348218

Adjacent sequences: A270637 A270638 A270639 * A270641 A270642 A270643

Keyword

nonn

Author

Michael Somos, Mar 20 2016

Status

approved