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%I #13 Mar 30 2012 18:37:31
%S 1,1,0,-1,-1,-1,-2,-3,-3,-2,0,2,2,0,-1,2,8,12,11,8,7,7,5,2,1,2,4,7,7,
%T -3,-21,-34,-34,-28,-28,-37,-46,-42,-22,-1,-1,-28,-62,-75,-60,-35,-16,
%U 1,25,53,77,93,97,90,91,121,165,175,129,70,64,127,213,267,273,261,278,329,340,225,11,-155,-160,-50,25,-40,-223,-406,-475
%N The q-exponential of x, E_q(x,q), evaluated at q=-x.
%C This q-exponential of x is defined by:
%C E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * x^n/faq(n,q),
%C where
%C log(E_q(x,q)) = Sum_{n>=1} (q-1)^n/(q^n-1) * x^n/n,
%C and faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
%C See A152398 for the dual q-exponential function.
%H Paul D. Hanna, <a href="/A198197/b198197.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a> from MathWorld.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a> from MathWorld.
%F G.f.: E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product_{k=1..n} (1 - (-x)^k)/(1+x).
%F G.f.: E_q(x,-x) = exp( Sum_{n>=1} -(1+x)^n/(1-(-x)^n) * (-x)^n/n ).
%F G.f.: E_q(x,-x) = Product_{n>=1} (1 - (1+x)*(-x)^n).
%e G.f.: E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 +...
%e where
%e E_q(x,-x) = 1 + x - x^3/(1-x) - x^6/((1-x)*(1-x+x^2)) + x^10/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)) + x^15/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)*(1-x+x^2-x^3+x^4)) +...
%e The g.f. equals the product:
%e E_q(x,-x) = (1 + (1+x)*x) * (1 - (1+x)*x^2) * (1 + (1+x)*x^3) * (1 - (1+x)*x^4) * (1 + (1+x)*x^5) * (1 - (1+x)*x^6) *...
%e The logarithm of the g.f. equals the series:
%e log(E_q(x,-x)) = x - (1+x)^2/(1-x^2)*x^2/2 + (1+x)^3/(1+x^3)*x^3/3 - (1+x)^4/(1-x^4)*x^4/4 + (1+x)^5/(1+x^5)*x^5/5 - (1+x)^6/(1-x^6)*x^6/6 +...
%e more explicitly,
%e log(E_q(x,-x)) = x - x^2/2 - 2*x^3/3 - x^4/4 - 4*x^5/5 - 10*x^6/6 - 13*x^7/7 - 17*x^8/8 - 20*x^9/9 - 16*x^10/10 +...
%o (PARI) {a(n)=local(E_q=sum(k=0, n, (-x)^(k*(k-1)/2)*x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))));polcoeff(E_q, n)}
%o (PARI) {a(n)=local(q=-x,E_q=exp(sum(k=1, n, (q-1)^k/(q^k-1) * x^k/k)+x*O(x^n)));polcoeff(E_q, n)}
%o (PARI) {a(n)=polcoeff(prod(k=1, n, 1-(1+x)*(-x)^k+x*O(x^n)), n)}
%Y Cf. A198262 (log), A152398 (e_q), A198199, A198200.
%K sign
%O 0,7
%A _Paul D. Hanna_, Oct 23 2011
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