A235870 Expansion of ( f(-q)^12 + 22 * q * f(-q)^6 * f(-q^5)^6 + 125 * q^2 * f(-q^5)^12 ) / (f(-q) * f(-q^5))^2 in powers of q where f() is a Ramanujan theta function.

1, 12, 72, 264, 696, 1380, 2304, 3192, 5400, 6924, 12600, 12384, 18912, 20184, 28512, 39000, 43032, 45432, 63144, 63600, 101640, 88944, 110304, 112104, 151200, 174540, 183024, 188400, 231936, 225000, 351360, 274704, 346392, 344448, 407952, 479400, 509592

Offset: 0

Author

Michael Somos, Jun 13 2014

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 12*q + 72*q^2 + 264*q^3 + 696*q^4 + 1380*q^5 + 2304*q^6 + ...
		

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. A028887.

Programs

Magma

A := Basis( ModularForms( Gamma0(5), 4), 36); A[1] + 12*A[2] + 72*A[3]; /* Michael Somos, Jun 13 2014 */

PARI

{a(n) = my(A, u1, u5); if( n<0, 0, A = x * O(x^n); u1 = eta(x + A); u5 = eta(x^5 + A); polcoeff( ( u1^12 + 22*x * (u1 * u5)^6 + 125*x^2 * u5^12 ) / (u1 * u5)^2, n))};

PARI

{a(n) = my(A, v1, v3); if( n<0, 0, A = x * O(x^n); v1 = eta(x + A) * eta(x^5 + A) ; v3 = eta(x^3 + A) * eta(x^15 + A) ; polcoeff( ( v1^4 + 9*x * (v1 * v3)^2 + 27*x^2 * v3^4 )^2 / (v1 * v3)^2, n))};

Sage

A = ModularForms( Gamma0(5), 4, prec=36) . basis(); A[1] + 12/13 * (3*A[0] + 10*A[2]); # Michael Somos, Jun 13 2014

Formula

Expansion of ( ( (f(-q) * f(-q^5))^4 + 9*q * (f(-q) * f(-q^3) * f(-q^5) * f(-q^15))^2 + 27*q * (f(-q^3) * f(-q^15))^4 ) / (f(-q) * f(-q^3) * f(-q^5) * f(-q^15)) )^2 in powers of q where f() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (t/i)^4 f(t) where q = exp(2 Pi i t).

Convolution square of A028887.