A152399 - OEIS

%I #3 Mar 30 2012 18:37:15

%S 1,1,4,9,16,22,29,49,94,156,221,318,521,883,1429,2257,3605,5836,9463,

%T 15264,24539,39579,64148,103990,168141,271623,439276,711055,1150750,

%U 1861287,3010318,4870449,7881944,12754455,20635589,33385764,54018447

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

%C The g.f.s for this sequence illustrates 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 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 L.g.f.: log(e_q(x,-x)) = log(Sum_{n>=0} x^n/[Product_{k=1..n} (1-(-x)^k)/(1+x)]).

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

%e L.g.f.: 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 +...

%e e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 17*x^7 +... (A152398).

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

%Y Cf. A152398 (e_q(x, -x)).

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 16 2008