A210656 Expansion of psi(x^3) * phi(-x)^2 / phi(-x^2) in power of x where phi(), psi() are Ramanujan theta functions.

1, 8, 36, 130, 412, 1176, 3105, 7712, 18192, 41098, 89476, 188592, 386322, 771528, 1506036, 2879688, 5403628, 9966408, 18092599, 32366288, 57117660, 99526362, 171378512, 291841464, 491812740, 820684904, 1356794820, 2223458146, 3613417008, 5825889936

Offset: 0

Author

Michael Somos, Mar 27 2012

Keywords

• nonn

Comments

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

Examples

			1 + 8*x + 36*x^2 + 130*x^3 + 412*x^4 + 1176*x^5 + 3105*x^6 + 7712*x^7 + ...
q^3 + 8*q^7 + 36*q^11 + 130*q^15 + 412*q^19 + 1176*q^23 + 3105*q^27 + ...
		

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

Crossrefs

Cf. A001936.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 - x^(2*k))^4 * (1 - x^(6*k))^2 / ((1 - x^k)^4 * (1 - x^(3*k)) * (1 - x^(4*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^4 * eta(x^3 + A) * eta(x^ 4 + A)) )^2, n))}

Formula

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

Euler transform of period 12 sequence [ 8, 0, 10, 2, 8, -2, 8, 2, 10, 0, 8, 0, ...].

A001936(9*n + 2) - A001936(n) = 4 * a(3*n). A001936(9*n + 5) = 4 * a(3*n + 1). A001936(9*n + 8) = 4 * a(3*n + 2).

a(n) ~ exp(sqrt(3*n)*Pi) / (32*sqrt(2)*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017