A193442 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.

1, 2, 24, 624, 27744, 1857600, 173256192, 21357471744, 3350185712640, 649812612225024, 152385461527633920, 42429768718712094720, 13819620038445598408704, 5199913478124022299033600, 2236448840442614178722611200, 1089467246881095674146487009280

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..15.

Formula

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

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.

Example

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)! +...
where
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) +...

Prog

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

Crossrefs

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

Sequence in context: A210909 A009251 A009447 * A123741 A302579 A012216

Adjacent sequences: A193439 A193440 A193441 * A193443 A193444 A193445

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2011

Status

approved