A230277 Expansion of f(-q^6)^6 * ( q * chi(-q^3)^4 + 2*q^2 / chi(-q^3)^4 ) in powers of q where chi(), f() are Ramanujan theta functions.

1, 2, 0, -4, 8, 0, 0, 8, 0, 16, 0, 0, -10, 0, 0, -16, -16, 0, 0, -32, 0, 0, 0, 0, 39, -20, 0, 0, -40, 0, 0, 32, 0, -32, 0, 0, -70, 0, 0, 64, 80, 0, 0, 0, 0, 0, 0, 0, 49, 78, 0, 40, 56, 0, 0, 0, 0, -80, 0, 0, -22, 0, 0, -64, -80, 0, 0, 64, 0, 0, 0, 0, 110

Offset

1,2

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 = 1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q^3)^2 * eta(q^6))^2 + 2 * (eta(q^6)^5 / eta(q^3)^2)^2 in powers of q.

a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(3^e) = 0^e, a(p^e) = p^e * (1 + (-1)^e)/2 if p == 3, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) and a(p) = t * (-1)^(t mod 3) if p == 1, 5 (mod 8) where t = 2 * if( p == 5 (mod 6) then 4*x*y else p - 8*y^2 ) and p = x^2 + 4*y^2.

a(2*n) = -2 * a(n). a(3*n) = a(4*n + 3) = 0.

a(3*n + 1) = A230280(n). a(3*n + 2) = 2 * A230278(n).

Example

G.f. = q + 2*q^2 - 4*q^4 + 8*q^5 + 8*q^8 + 16*q^10 - 10*q^13 - 16*q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ With[{t = QPochhammer[ q^3, q^6]^4}, QPochhammer[ q^6]^6 (q t + 2 q^2 / t)], {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A)^2 * eta(x^6 + A))^2 + 2 * x * (eta(x^6 + A)^5 / eta(x^3 + A)^2)^2, n))}
(PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor( n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==3, 0, if( p==2, -(-2)^e, if( p%8==3 || p%8==7, if(e%2, 0, p^e), for( i = 1, sqrtint( p\4), if( issquare( p - 4*i^2, &x), y = i; break)); y = 2 * if( p%6 == 5, 4*x*y, p - 8*y^2); a1 = y = if( y%3==2, y, -y); a0 = 1; for( i=2, e, x = y*a1 - p^2*a0; a0 = a1; a1 = x); a1))))))}

Crossrefs

Cf. A230278, A230280.

Sequence in context: A344769 A011121 A255643 * A117902 A021087 A120558

Adjacent sequences: A230274 A230275 A230276 * A230278 A230279 A230280

Keyword

sign,mult

Author

Michael Somos, Oct 15 2013

Status

approved