A198262 Logarithmic derivative of the q-exponential of x, E_q(x,q), evaluated at q=-x.

1, -1, -2, -1, -4, -10, -13, -17, -20, -16, -21, -46, -77, -99, -97, -81, -101, -172, -265, -376, -499, -595, -666, -806, -1129, -1639, -2234, -2871, -3624, -4615, -6044, -8177, -11178, -15063, -19793, -25444, -32633, -42751, -57410, -77712, -104221, -137679, -180126, -235775, -311020, -413748

Offset

0,3

Links

Table of n, a(n) for n=0..45.

Formula

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

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

L.g.f.: log(E_q(x,-x)), where E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product_{k=1..n} (1 - (-x)^k)/(1+x)).

Example

L.g.f.: 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 +...
The logarithm of E_q(x,-x) 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 +...
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)) +...
more explicitly,
E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 +...+ A198197(n)*x^n +...

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)))); n*polcoeff(log(E_q), n)}
(PARI) {a(n)=local(Lgf=sum(k=1, n, -(1+x)^k/(1-(-x)^k) * (-x)^k/k)+x*O(x^n)); n*polcoeff(Lgf, n)}
(PARI) {a(n)=local(Lgf=sum(k=1, n, log(1-(1+x)*(-x)^k+x*O(x^n)))); n*polcoeff(Lgf, n)}

Crossrefs

Cf. A198197.

Sequence in context: A329709 A038521 A134654 * A085421 A001071 A121198

Adjacent sequences: A198259 A198260 A198261 * A198263 A198264 A198265

Keyword

sign

Author

Paul D. Hanna, Oct 23 2011

Status

approved