|
Search: id:A152398
|
|
|
| A152398 |
|
The q-exponential of x, e_q(x,q), evaluated at q=-x. |
|
+0
2
|
|
| 1, 1, 1, 2, 4, 7, 11, 17, 28, 48, 80, 128, 204, 332, 545, 887, 1432, 2313, 3750, 6086, 9859, 15944, 25788, 41749, 67604, 109415, 177017, 286409, 463495, 750081, 1213713, 1963771, 3177444, 5141446, 8319390, 13461189, 21780519, 35241682
(list; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
The g.f.s for this sequence illustrates 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
|
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. [From Vladeta Jovovic (vladeta(AT)eunet.yu), 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... [From Paul D. Hanna (pauldhanna(AT)juno.com), Dec 20 2008]
|
|
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).
|
|
PROGRAM
|
(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Dec 20 2008]
|
|
CROSSREFS
|
Cf. A152399 (log(e_q(x, -x))).
Sequence in context: A023429 A023428 A093911 this_sequence A023427 A129929 A073738
Adjacent sequences: A152395 A152396 A152397 this_sequence A152399 A152400 A152401
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Dec 16 2008
|
|
|
Search completed in 0.005 seconds
|
