A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).

1, 2, 2, 0, -2, 0, 4, 4, -2, -6, 0, 8, 4, -8, -8, 8, 14, -4, -18, 0, 24, 8, -28, -20, 28, 34, -24, -48, 16, 64, 0, -76, -18, 88, 44, -96, -78, 96, 116, -88, -160, 68, 208, -32, -252, -16, 296, 84, -332, -170, 354, 272, -360, -392, 344, 528, -296, -672, 216, 824, -96, -976, -72, 1116, 286, -1240, -552, 1336, 876, -1384

Offset: 0

Author

Joerg Arndt, Aug 07 2011

Keywords

• sign

Comments

Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).

Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.

Examples

			1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• J. Arndt, Regularized q-exponential and q-trigonometric series of the first kind

Crossrefs

Cf. A015128 E(+q,+q), A002448 E(+q,-q), A000122 E(-q,+q), A004402 E(-q,-q), A080054 E(+q^2,+q), A108494 E(+q^2,-q), A300574, A300575.

Programs

PARI

N=66; q='q+O('q^N); /* that many terms */
gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
Vec(gf) /* show terms */
/* Alternative computation of the g.f. using a product form */
V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );

PARI

{a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n) )^[ 0, -2, 1, 2, 0, -2, -1, 2][k%8 + 1]), n))} /* Michael Somos, Feb 26 2012 */

Formula

Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - Michael Somos, Feb 26 2012

G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).

G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).

G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k)/( prod(k=1..n, 1-(-q^2)^k) * prod(k=0..n-1, 1-q*(-q^2)^k ) ) * q^n * (-q^2)^(n*(n-1)/2) ).

Convolution of A300574 and A300575. - Seiichi Manyama, Nov 22 2019