A198197 The q-exponential of x, E_q(x,q), evaluated at q=-x.

1, 1, 0, -1, -1, -1, -2, -3, -3, -2, 0, 2, 2, 0, -1, 2, 8, 12, 11, 8, 7, 7, 5, 2, 1, 2, 4, 7, 7, -3, -21, -34, -34, -28, -28, -37, -46, -42, -22, -1, -1, -28, -62, -75, -60, -35, -16, 1, 25, 53, 77, 93, 97, 90, 91, 121, 165, 175, 129, 70, 64, 127, 213, 267, 273, 261, 278, 329, 340, 225, 11, -155, -160, -50, 25, -40, -223, -406, -475

Offset

0,7

Comments

This q-exponential of x is defined by:

E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * x^n/faq(n,q),

where

log(E_q(x,q)) = Sum_{n>=1} (q-1)^n/(q^n-1) * x^n/n,

and faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.

See A152398 for the dual q-exponential function.

Links

Paul D. Hanna, 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*(n-1)/2) * x^n/Product_{k=1..n} (1 - (-x)^k)/(1+x).

G.f.: E_q(x,-x) = exp( Sum_{n>=1} -(1+x)^n/(1-(-x)^n) * (-x)^n/n ).

G.f.: E_q(x,-x) = Product_{n>=1} (1 - (1+x)*(-x)^n).

Example

G.f.: E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 +...
where
E_q(x,-x) = 1 + x - x^3/(1-x) - x^6/((1-x)*(1-x+x^2)) + x^10/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)) + x^15/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)*(1-x+x^2-x^3+x^4)) +...
The g.f. equals the product:
E_q(x,-x) = (1 + (1+x)*x) * (1 - (1+x)*x^2) * (1 + (1+x)*x^3) * (1 - (1+x)*x^4) * (1 + (1+x)*x^5) * (1 - (1+x)*x^6) *...
The logarithm of the g.f. equals the series:
log(E_q(x,-x)) = x - (1+x)^2/(1-x^2)*x^2/2 + (1+x)^3/(1+x^3)*x^3/3 - (1+x)^4/(1-x^4)*x^4/4 + (1+x)^5/(1+x^5)*x^5/5 - (1+x)^6/(1-x^6)*x^6/6 +...
more explicitly,
log(E_q(x,-x)) = x - x^2/2 - 2*x^3/3 - x^4/4 - 4*x^5/5 - 10*x^6/6 - 13*x^7/7 - 17*x^8/8 - 20*x^9/9 - 16*x^10/10 +...

Prog

(PARI) {a(n)=local(E_q=sum(k=0, n, (-x)^(k*(k-1)/2)*x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n)))); polcoeff(E_q, n)}
(PARI) {a(n)=local(q=-x, E_q=exp(sum(k=1, n, (q-1)^k/(q^k-1) * x^k/k)+x*O(x^n))); polcoeff(E_q, n)}
(PARI) {a(n)=polcoeff(prod(k=1, n, 1-(1+x)*(-x)^k+x*O(x^n)), n)}

Crossrefs

Cf. A198262 (log), A152398 (e_q), A198199, A198200.

Sequence in context: A099028 A357521 A279645 * A203400 A077869 A076585

Adjacent sequences: A198194 A198195 A198196 * A198198 A198199 A198200

Keyword

sign

Author

Paul D. Hanna, Oct 23 2011

Status

approved