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

0,2

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^(-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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A001936.

Sequence in context: A014477 A034998 A121255 * A119767 A024208 A000427

Adjacent sequences: A210653 A210654 A210655 * A210657 A210658 A210659

Keyword

nonn

Author

Michael Somos, Mar 27 2012

Status

approved