A136599 Expansion of (eta(q) * eta(q^15))^3 in powers of q.

1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -14, 9, 0, -15, 0, 0, 34, 0, 0, 0, -27, 0, 0, -15, 0, 33, 0, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 0, 45, 0, -14, -15, 0, 25, 0, 0, -86, 0, 0, 0, 66, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 42, 0, 0, 0, -63, 0, 0, -75, 0, 0, 0, 0

Offset

2,2

Links

Table of n, a(n) for n=2..79.

Formula

Euler transform of period 15 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

a(n) nonzero or n=0 if and only if n is in A028955.

G.f.: x^2 * (Product_{k>0} (1 - x^k) * (1 - x^(15*k)))^3.

a(3*n) = -3 * A030220(n). a(3*n + 1) = 0. - Michael Somos, Oct 13 2015

A115155(n) = a(n) + A030220(n). - Michael Somos, Oct 13 2015

Example

G.f. = q^2 - 3*q^3 + 5*q^5 - 7*q^8 + 9*q^12 - 14*q^17 + 9*q^18 - 15*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* Michael Somos, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^15 + A))^3, n))};
(Magma) A := Basis( CuspForms( Gamma1(15), 3), 80); A[2] - 3*A[3] + 5*A[5] - 7*A[8]; /* Michael Somos, Oct 13 2015 */

Crossrefs

Cf. A028955, A030220, A115155.

Sequence in context: A010816 A133089 A198954 * A227498 A131986 A002656

Adjacent sequences: A136596 A136597 A136598 * A136600 A136601 A136602

Keyword

sign

Author

Michael Somos, Jan 11 2008

Status

approved