%I #17 Aug 16 2018 03:08:22

%S 1,2,5,24,129,960,7965,80640,903105,11612160,163451925,2554675200,

%T 43259364225,797058662400,15764670046125,334764638208000,

%U 7571150452490625,182111963185152000,4634731528895593125,124564582818643968000

%N E.g.f.: -log( sqrt(1-x^2) - x ).

%C Compare e.g.f. to arccosh(x) = log(sqrt(x^2-1) + x).

%F a(2*n) = 2^n*(2*n-1)! for n>=1.

%F a(n) = A100097(n+1)*(n-1)!/2^n for n>=1.

%F a(n) = (n-1)!/2^n * Sum_{k=0..floor((n+1)/2)} C(n+1,k)*A000129(n+1-2*k) for n >= 1. [From a formula of Paul Barry in A100097]

%F E.g.f.: log( (sqrt(1-x^2) + x)/(1-2*x^2) ).

%e E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 24*x^4/4! + 129*x^5/5! + ...

%e where

%e exp(A(x)) = 1 + 2*(x/2) + 6*(x/2)^2 + 16*(x/2)^3 + 46*(x/2)^4 + 128*(x/2)^5 + ... + A098617(n)*(x/2)^n + ...

%t With[{nn=30},Rest[CoefficientList[Series[-Log[Sqrt[1-x^2]-x],{x,0,nn}], x] Range[0,nn]!]] (* _Harvey P. Dale_, Dec 01 2011 *)

%o (PARI) {a(n)=n!*polcoeff(-log(sqrt(1-x^2+x*O(x^n))-x),n)}

%o (PARI) {A000129(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}

%o {a(n)=if(n<1,0,sum(k=0,floor((n+1)/2),binomial(n+1, k)*A000129(n+1-2*k))*(n-1)!/2^n)}

%Y Cf. A098617, A100097.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 21 2011