cp's OEIS Frontend

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

%I A198262 #11 Feb 16 2025 04:39:37
%S A198262 1,-1,-2,-1,-4,-10,-13,-17,-20,-16,-21,-46,-77,-99,-97,-81,-101,-172,
%T A198262 -265,-376,-499,-595,-666,-806,-1129,-1639,-2234,-2871,-3624,-4615,
%U A198262 -6044,-8177,-11178,-15063,-19793,-25444,-32633,-42751,-57410,-77712,-104221,-137679,-180126,-235775,-311020,-413748
%N A198262 Logarithmic derivative of the q-exponential of x, E_q(x,q), evaluated at q=-x.
%F A198262 L.g.f.: log(E_q(x,-x)) = Sum_{n>=1} -(1+x)^n/(1-(-x)^n) * (-x)^n/n.
%F A198262 L.g.f.: log(E_q(x,-x)) = Sum_{n>=1} log(1 - (1+x)*(-x)^n).
%F A198262 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).
%e A198262 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 +...
%e A198262 The logarithm of E_q(x,-x) equals the series:
%e A198262 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 +...
%e A198262 where
%e A198262 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)) +...
%e A198262 more explicitly,
%e A198262 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 +...
%o A198262 (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)}
%o A198262 (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)}
%o A198262 (PARI) {a(n)=local(Lgf=sum(k=1, n, log(1-(1+x)*(-x)^k+x*O(x^n))));n*polcoeff(Lgf, n)}
%Y A198262 Cf. A198197.
%K A198262 sign
%O A198262 0,3
%A A198262 _Paul D. Hanna_, Oct 23 2011