A258090 Expansion of q^(-5/6) * (eta(q) * eta(q^6)^2 / eta(q^3))^2 in powers of q.

1, -2, -1, 4, -3, 0, 3, 0, 1, -2, -2, -4, 0, 2, 3, -4, 9, 6, -9, 0, -6, 2, 3, 4, -7, 8, 0, -12, -3, -6, 6, 0, 9, 0, 8, 4, 2, -6, -5, 8, -7, -10, -1, 4, 5, 2, -13, 0, 9, -8, -2, 12, -3, 4, 0, -4, -16, 6, -1, 12, 10, 0, 6, 0, 1, -8, 15, -12, 0, -6, 1, -16, -16

Offset: 0

Author

Michael Somos, May 19 2015

Keywords

• sign

Examples

			G.f. = 1 - 2*x - x^2 + 4*x^3 - 3*x^4 + 3*x^6 + x^8 - 2*x^9 - 2*x^10 + ...
G.f. = q^5 - 2*q^11 - q^17 + 4*q^23 - 3*q^29 + 3*q^41 + q^53 - 2*q^59 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A030188.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3])^2, {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2 / eta(x^3 + A))^2, n))};

PARI

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 5; A = factor(n); -1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, (-1)^e, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( x^3 - x^2 - 4*x + 4, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))};

PARI

q='q+O('q^99); Vec((eta(q)*eta(q^6)^2/eta(q^3))^2) \\ Altug Alkan, Aug 02 2018

Formula

Euler transform of period 6 sequence [ -2, -2, 0, -2, -2, -4, ...].

G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 + x^(3*k))^4.

-2 * a(n) = A030188(3*n + 2).