A193442 - OEIS

%I #13 Mar 30 2012 18:37:28

%S 1,2,24,624,27744,1857600,173256192,21357471744,3350185712640,

%T 649812612225024,152385461527633920,42429768718712094720,

%U 13819620038445598408704,5199913478124022299033600,2236448840442614178722611200,1089467246881095674146487009280

%N E.g.f.: exp( Sum_{n>=1} x^(2*n)/A000108(n) ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where A000108 is the Catalan numbers.

%C Sum_{n>=0} a(n)/(2*n)! = exp(1 + 4*sqrt(3)*Pi/27) = 6.08686426907670...

%F E.g.f.: exp(L(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!,

%F where L(x) = -1 + 2*(8+x^2)/(4-x^2)^2 + 24*x*atan(x/sqrt(4-x^2))/sqrt((4-x^2)^5) from a formula given in A121839.

%e E.g.f.: A(x) = 1 + 2*x^2/2! + 24*x^4/4! + 624*x^6/6! + 27744*x^8/8! + 1857600*x^10/10! + 173256192*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...

%e where

%e log(A(x)) = x^2 + x^4/2 + x^6/5 + x^8/14 + x^10/42 + x^12/132 + x^14/429 + x^16/1430 +...+ (n+1)*x^(2*n)/C(2*n,n) +...

%o (PARI) {a(n)=(2*n)!*polcoeff(exp(sum(m=1,n,(m+1)*x^(2*m)/binomial(2*m,m))+O(x^(2*n+1))),2*n)}

%o (PARI) /* Using formula for e.g.f. = exp(L(x)): */

%o {a(n)=local(Ox=O(x^(2*n+1)), L=-1 + 2*(8+x^2)/(4-x^2 +Ox)^2 + 24*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^5)); (2*n)!*polcoeff(exp(L), 2*n)}

%Y Cf. A193441, A193443, A193444, A000108 (Catalan), A121839.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 25 2011