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.

1, 1, 9, 177, 6081, 320625, 23901993, 2382903873, 305213701185, 48729724204833, 9471295552801545, 2198860046959656465, 600311814859681301889, 190227653770262659729425, 69194247433728324962214825, 28616922449430718198313413665, 13345389352004839017903164910465

Offset

0,3

Comments

Sum_{n>=0} a(n)/(2*n)! = exp(1/2 + 2*sqrt(3)*Pi/27) = 2.4671571229001...

Links

Table of n, a(n) for n=0..16.

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 + 12*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 + 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)! +...
where
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)) +...

Prog

(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)}
(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 + 12*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^5)); (2*n)!*polcoeff(exp(L), 2*n)}

Crossrefs

Cf. A193441, A193442, A000108 (Catalan), A121839.

Sequence in context: A141363 A157774 A232694 * A003711 A009009 A220267

Adjacent sequences: A193440 A193441 A193442 * A193444 A193445 A193446

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2011

Status

approved