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

%I A193863 #29 Nov 22 2019 06:26:51
%S A193863 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,
%T A193863 -24,-48,16,64,0,-76,-18,88,44,-96,-78,96,116,-88,-160,68,208,-32,
%U A193863 -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
%N A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).
%C A193863 Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).
%C A193863 Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.
%H A193863 Seiichi Manyama, <a href="/A193863/b193863.txt">Table of n, a(n) for n = 0..1000</a>
%H A193863 J. Arndt, <a href="http://www.jjj.de/pari/qtrig1r.gpi">Regularized q-exponential and q-trigonometric series of the first kind</a>
%F A193863 Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - _Michael Somos_, Feb 26 2012
%F A193863 G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).
%F A193863 G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).
%F A193863 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) ).
%F A193863 Convolution of A300574 and A300575. - _Seiichi Manyama_, Nov 22 2019
%e A193863 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...
%o A193863 (PARI) N=66; q='q+O('q^N); /* that many terms */
%o A193863 gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
%o A193863 Vec(gf) /* show terms */
%o A193863 /* Alternative computation of the g.f. using a product form */
%o A193863 V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
%o A193863 gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );
%o A193863 (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 */
%Y A193863 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.
%K A193863 sign
%O A193863 0,2
%A A193863 _Joerg Arndt_, Aug 07 2011