A259033 Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.

1, 6, 23, 76, 221, 584, 1443, 3368, 7497, 16046, 33190, 66628, 130288, 248858, 465387, 853836, 1539425, 2731462, 4775703, 8236856, 14027754, 23609794, 39301171, 64747876, 105638153, 170778512, 273704800, 435079524, 686237877, 1074405242, 1670333294, 2579418528

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 6 sequence [ 6, 2, 8, 2, 6, 0, ...].

a(n) = A263528(3*n + 2). -2 * a(n) = A261369(2*n + 1) = A213265(6*n + 5) = A262930(6*n + 5).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 + 6*x + 23*x^2 + 76*x^3 + 221*x^4 + 584*x^5 + 1443*x^6 + 3368*x^7 + ...
G.f. = q^5 + 6*q^11 + 23*q^17 + 76*q^23 + 221*q^29 + 584*q^35 + 1443*q^41 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/4) x^(-3/4) EllipticTheta[ 2, 0, x^(3/2)]^2 QPochhammer[ x^2]^4 / QPochhammer[ x]^6, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[((1 + x^k)^2 * (1 + x^(3*k))^2 * (1 - x^(3*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)

Prog

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

Crossrefs

Cf. A213265, A261369, A262930, A263528.

Sequence in context: A038737 A038797 A136530 * A374856 A054459 A060188

Adjacent sequences: A259030 A259031 A259032 * A259034 A259035 A259036

Keyword

nonn

Author

Michael Somos, Nov 07 2015

Status

approved