A113277 Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.

1, 2, 0, 0, 0, -4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(2^e) = -(-2)^(e/2), a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).

Euler transform of period 2 sequence [ 2, -3, ...].

G.f.: Sum_{k} (3*k + 1) * (-x)^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5.

Expansion of psi(x^2) * f(x)^2 = phi(x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3456^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A080332.

a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(25*n + 8) = -5 * a(n). A114855(n) = (-1)^n * a(n). a(4*n + 1) = 2 * A114855(n). a(8*n) = A080332(n).

Example

1 + 2*x - 4*x^5 - 5*x^8 + 7*x^16 + 8*x^21 - 10*x^33 - 11*x^40 + 13*x^56 + ...
q + 2*q^4 - 4*q^16 - 5*q^25 + 7*q^49 + 8*q^64 - 10*q^100 - 11*q^121 +...

Mathematica

QP = QPochhammer; s = QP[q^2]^5/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x + A)^2, n))}
(PARI) {a(n) = if(issquare( 3*n + 1, &n), n * (-1)^(n%3 + n), 0)}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod (k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( e%2, 0, (-1)^(p==2) * (-(-1)^(p%3) * p)^(e/2)))))}

Crossrefs

Cf. A080332, A114855.

Sequence in context: A262162 A246950 A204531 * A114855 A221381 A100951

Adjacent sequences: A113274 A113275 A113276 * A113278 A113279 A113280

Keyword

sign

Author

Michael Somos, Oct 21 2005

Status

approved