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A152398 The q-exponential of x, e_q(x,q), evaluated at q = -x.

%I A152398 #16 Feb 16 2025 08:33:09
%S A152398 1,1,1,2,4,7,11,17,28,48,80,128,204,332,545,887,1432,2313,3750,6086,
%T A152398 9859,15944,25788,41749,67604,109415,177017,286409,463495,750081,
%U A152398 1213713,1963771,3177444,5141446,8319390,13461189,21780519,35241682
%N A152398 The q-exponential of x, e_q(x,q), evaluated at q = -x.
%C A152398 The g.f.s for this sequence illustrate the following formula:
%C A152398 log(e_q(x,q)) = Sum_{n>=1} (1-q)^n/(1-q^n)*x^n/n, where
%C A152398 e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential of x and
%C A152398 faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
%H A152398 Seiichi Manyama, <a href="/A152398/b152398.txt">Table of n, a(n) for n = 0..1000</a>
%H A152398 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a> from MathWorld.
%H A152398 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a> from MathWorld.
%F A152398 G.f.: e_q(x,-x) = Sum_{n>=0} x^n/(Product_{k=1..n} (1-(-x)^k)/(1+x)).
%F A152398 G.f.: e_q(x,-x) = exp( Sum_{n>=1} x^n*(1+x)^n/(1-(-x)^n)/n ).
%F A152398 G.f.: 1/Product_{k>0} 1+(1+x)*(-x)^k. - _Vladeta Jovovic_, Dec 19 2008
%F A152398 a(n) ~ c/r^n where r = (sqrt(5) - 1)/2 = 0.6180339887... and c = 0.652419554233497352459208493304650..., where e_q(-r,r) = 0.887276226980250304353751667447441... - _Paul D. Hanna_, Dec 20 2008
%F A152398 c = 1 / (r * sqrt(5) * QPochhammer((1-sqrt(5))/2)). - _Vaclav Kotesovec_, Oct 22 2020
%e A152398 G.f.: e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + ...
%e A152398 log(e_q(x,-x)) = x + x^2/2 + 4*x^3/3 + 9*x^4/4 + 16*x^5/5 + 22*x^6/6 + ... (A152399).
%o A152398 (PARI) a(n)=polcoeff(sum(k=0,n,x^k/(prod(j=1,k,(1-(-x)^j)/(1+x))+x*O(x^n))),n)
%o A152398 (PARI) a(n)=polcoeff(exp(sum(k=1,n,x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)),n)
%o A152398 (PARI) {a(n)=polcoeff(1/prod(k=1,n,1+(1+x)*(-x)^k+x*O(x^n)),n)} \\ _Paul D. Hanna_, Dec 20 2008
%Y A152398 Cf. A152399: log(e_q(x, -x)); A227681, A306749.
%K A152398 nonn
%O A152398 0,4
%A A152398 _Paul D. Hanna_, Dec 16 2008