A152399 Log of the q-exponential of x, e_q(x,q), evaluated at q=-x.

1, 1, 4, 9, 16, 22, 29, 49, 94, 156, 221, 318, 521, 883, 1429, 2257, 3605, 5836, 9463, 15264, 24539, 39579, 64148, 103990, 168141, 271623, 439276, 711055, 1150750, 1861287, 3010318, 4870449, 7881944, 12754455, 20635589, 33385764, 54018447

Offset

1,3

Comments

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

Table of n, a(n) for n=1..37.

Eric Weisstein, q-Exponential Function from MathWorld.

Eric Weisstein, q-Factorial from MathWorld.

Formula

L.g.f.: log(e_q(x,-x)) = log(Sum_{n>=0} x^n/[Product_{k=1..n} (1-(-x)^k)/(1+x)]).

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

Example

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

Prog

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

Crossrefs

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

Sequence in context: A313348 A313349 A313350 * A022822 A001639 A309138

Adjacent sequences: A152396 A152397 A152398 * A152400 A152401 A152402

Keyword

nonn

Author

Paul D. Hanna, Dec 16 2008

Status

approved