OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
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).
EXAMPLE
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 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3])^2, {x, 0, n}];
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 19 2015
STATUS
approved