A224916 Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan theta function.

1, 2, 7, 14, 31, 58, 112, 196, 347, 580, 966, 1554, 2485, 3872, 5993, 9102, 13719, 20384, 30068, 43836, 63481, 91048, 129763, 183448, 257839, 359862, 499583, 689312, 946416, 1292388, 1756838, 2376598, 3201557, 4293942, 5736736, 7633702, 10121408, 13370634

Offset: 0

Author

Michael Somos, Apr 19 2013

Keywords

• nonn

Comments

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

Examples

			1 + 2*x + 7*x^2 + 14*x^3 + 31*x^4 + 58*x^5 + 112*x^6 + 196*x^7 + 347*x^8 + ...
q^5 + 2*q^17 + 7*q^29 + 14*q^41 + 31*q^53 + 58*q^65 + 112*q^77 + 196*q^89 + ...
		

Links

• G. C. Greubel, 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

Crossrefs

Cf. A098613, A112160.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / (4 q^(1/2) QPochhammer[q]^2), {q, 0, n}]
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q^2, q^4]^4 / QPochhammer[ q, q^2]^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2]^4 - QPochhammer[ q, q^2]^4)/ 8, {q, 0, 2 n + 1}]

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)))^2, n))}

Formula

Expansion of q^(-5/12) * (eta(q^4)^2 / (eta(q) * eta(q^2)))^2 in powers of q.

Expansion of psi(x^2)^2 / f(-x)^2 = 1 / (chi(-x)^2 * chi(-x^2)^4) = 1 / (chi(x)^4 * chi(-x)^6 ) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Expansion of (chi(x)^4 - chi(-x)^4) / (8*x) in powers of x^2 where chi() is a Ramanujan theta function.

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

G.f.: Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k))^4.

G.f.: (Sum_{k>0} x^(k^2 - k)) / (Product_{k>0} (1 - x^k))^2. - Michael Somos, Jul 04 2013

a(n) = A112160(2*n + 1) / 4.

Convolution square of A098613. - Michael Somos, Jul 04 2013

a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015