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

0,2

Comments

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

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

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.

Example

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

Prog

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

Crossrefs

Cf. A028887.

Sequence in context: A188660 A047928 A300847 * A008533 A010024 A008414

Adjacent sequences: A235867 A235868 A235869 * A235871 A235872 A235873

Keyword

nonn

Author

Michael Somos, Jun 13 2014

Status

approved