A261203 Expansion of f(-x^6)^2 / (phi(-x) * phi(-x^9)) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 4, 8, 14, 24, 38, 60, 92, 140, 208, 304, 439, 626, 884, 1232, 1704, 2336, 3182, 4300, 5772, 7700, 10212, 13472, 17673, 23076, 29988, 38808, 50008, 64184, 82070, 104560, 132760, 167996, 211920, 266512, 334202, 417902, 521152, 648224, 804254, 995432

Offset: 0

Author

Michael Somos, Aug 11 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 60*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 14*q^9 + 24*q^11 + 38*q^13 + ...
		

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

Crossrefs

Cf. A261154, A261202.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^9]), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^9 + A)^2), n))};

Formula

Expansion of q^(-1/2) * eta(q^2) * eta(q^6)^2 * eta(q^18) / (eta(q)^2 * eta(q^9)^2) in powers of q.

Euler transform of period 18 sequence [ 2, 1, 2, 1, 2, -1, 2, 1, 4, 1, 2, -1, 2, 1, 2, 1, 2, 0, ...].

a(n) = A261154(2*n + 1).

Convolution inverse of A261202.

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017