A152398 - OEIS

%I #15 Oct 22 2020 04:12:05

%S 1,1,1,2,4,7,11,17,28,48,80,128,204,332,545,887,1432,2313,3750,6086,

%T 9859,15944,25788,41749,67604,109415,177017,286409,463495,750081,

%U 1213713,1963771,3177444,5141446,8319390,13461189,21780519,35241682

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

%C The g.f.s for this sequence illustrate the following formula:

%C log(e_q(x,q)) = Sum_{n>=1} (1-q)^n/(1-q^n)*x^n/n, where

%C e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential of x and

%C faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.

%H Seiichi Manyama, <a href="/A152398/b152398.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/(Product_{k=1..n} (1-(-x)^k)/(1+x)).

%F G.f.: e_q(x,-x) = exp( Sum_{n>=1} x^n*(1+x)^n/(1-(-x)^n)/n ).

%F G.f.: 1/Product_{k>0} 1+(1+x)*(-x)^k. - _Vladeta Jovovic_, Dec 19 2008

%F 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 c = 1 / (r * sqrt(5) * QPochhammer((1-sqrt(5))/2)). - _Vaclav Kotesovec_, Oct 22 2020

%e G.f.: e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + ...

%e 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 (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 (PARI) a(n)=polcoeff(exp(sum(k=1,n,x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)),n)

%o (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 Cf. A152399: log(e_q(x, -x)); A227681, A306749.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Dec 16 2008