cp's OEIS Frontend

A193443 E.g.f.: exp( Sum_{n>=1} x^(2*n)/(2*A000108(n)) ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where A000108 is the Catalan numbers.

%I A193443 #10 Mar 30 2012 18:37:28
%S A193443 1,1,9,177,6081,320625,23901993,2382903873,305213701185,
%T A193443 48729724204833,9471295552801545,2198860046959656465,
%U A193443 600311814859681301889,190227653770262659729425,69194247433728324962214825,28616922449430718198313413665,13345389352004839017903164910465
%N A193443 E.g.f.: exp( Sum_{n>=1} x^(2*n)/(2*A000108(n)) ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where A000108 is the Catalan numbers.
%C A193443  Sum_{n>=0} a(n)/(2*n)! = exp(1/2 + 2*sqrt(3)*Pi/27) = 2.4671571229001...
%F A193443  E.g.f.: exp(L(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!,
%F A193443 where L(x) = -1/2 + (8+x^2)/(4-x^2)^2 + 12*x*atan(x/sqrt(4-x^2))/sqrt((4-x^2)^5) from a formula given in A121839.
%e A193443 E.g.f.: A(x) = 1 + x^2/2! + 9*x^4/4! + 177*x^6/6! + 6081*x^8/8! + 320625*x^10/10! + 23901993*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
%e A193443 where
%e A193443 log(A(x)) = x^2/2 + x^4/4 + x^6/10 + x^8/28 + x^10/84 + x^12/264 + x^14/858 + x^16/2860 +...+ (n+1)*x^(2*n)/(2*C(2*n,n)) +...
%o A193443 (PARI) {a(n)=(2*n)!*polcoeff(exp(sum(m=1,n,(m+1)*x^(2*m)/binomial(2*m,m)/2)+O(x^(2*n+1))),2*n)}
%o A193443 (PARI) /* Using formula for e.g.f. = exp(L(x)): */
%o A193443 {a(n)=local(Ox=O(x^(2*n+1)),L=-1/2 + (8+x^2)/(4-x^2 +Ox)^2 + 12*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^5)); (2*n)!*polcoeff(exp(L),2*n)}
%Y A193443 Cf. A193441, A193442, A000108 (Catalan), A121839.
%K A193443 nonn
%O A193443 0,3
%A A193443 _Paul D. Hanna_, Jul 25 2011