A263051 Expansion of f(-x) * f(x^2, x^10) / f(-x^3)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 0, 1, -3, 1, 3, -5, 2, 6, -10, 4, 10, -18, 7, 17, -30, 12, 28, -49, 19, 44, -78, 31, 69, -120, 47, 105, -182, 71, 156, -271, 106, 229, -396, 154, 333, -572, 222, 475, -817, 317, 673, -1151, 445, 943, -1608, 620, 1307, -2226, 857, 1798, -3053, 1173, 2455

Offset

0,5

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-11/24) * eta(q) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2) * eta(q^3)^2 * eta(q^8) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [-1, 0, 1, -2, -1, 1, -1, -1, 1, 0, -1, 0, -1, 0, 1, -1, -1, 1, -1, -2, 1, 0, -1, 0, ...].

a(n) = - A137569(2*n + 1).

Example

G.f. = 1 - x + x^3 - 3*x^4 + x^5 + 3*x^6 - 5*x^7 + 2*x^8 + 6*x^9 + ...
G.f. = q^11 - q^35 + q^83 - 3*q^107 + q^131 + 3*q^155 - 5*q^179 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 2, Pi/4, x^3] QPochhammer[ -x^2, x^4] QPochhammer[ x] / QPochhammer[ x^3]^2, {x, 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)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)*eta(q^4)^2*eta(q^6)*eta(q^24)/(eta(q^2)*eta(q^3)^2*eta(q^8)*eta(q^12))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Cf. A137569.

Sequence in context: A225598 A126637 A110091 * A284130 A005474 A012264

Adjacent sequences: A263048 A263049 A263050 * A263052 A263053 A263054

Keyword

sign

Author

Michael Somos, Oct 08 2015

Status

approved