A152398 The q-exponential of x, e_q(x,q), evaluated at q = -x.

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

Offset: 0

Author

Paul D. Hanna, Dec 16 2008

Keywords

• nonn

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.

Examples

			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).
		

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.

Crossrefs

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

Programs

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

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