OFFSET
0,4
COMMENTS
The g.f.s for this sequence illustrate the following formula:
log(e_q(x,q)) = Sum_{n>=1} (1-q)^n/(1-q^n)*x^n/n, where
e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential of x and
faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein, q-Exponential Function from MathWorld.
Eric Weisstein, q-Factorial from MathWorld.
FORMULA
G.f.: e_q(x,-x) = Sum_{n>=0} x^n/(Product_{k=1..n} (1-(-x)^k)/(1+x)).
G.f.: e_q(x,-x) = exp( Sum_{n>=1} x^n*(1+x)^n/(1-(-x)^n)/n ).
G.f.: 1/Product_{k>0} 1+(1+x)*(-x)^k. - Vladeta Jovovic, Dec 19 2008
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
c = 1 / (r * sqrt(5) * QPochhammer((1-sqrt(5))/2)). - Vaclav Kotesovec, Oct 22 2020
EXAMPLE
G.f.: e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + ...
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).
PROG
(PARI) a(n)=polcoeff(sum(k=0, n, x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))), n)
(PARI) a(n)=polcoeff(exp(sum(k=1, n, x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)), n)
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 16 2008
STATUS
approved