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
Keywords
Examples
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)) +...
Programs
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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)}
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.
Comments