A193441 E.g.f.: exp( Sum_{n>=1} n!^2*x^(2*n)/(2*n)! ) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.

1, 1, 7, 111, 3089, 131985, 7977687, 645443295, 67165412385, 8722553971041, 1380689271177255, 261365482010524815, 58252017195624969009, 15086874107373899187825, 4490370671139664566269175, 1521257907398602231501780095, 581762614758928225569542394945

Offset

0,3

Comments

Sum_{n>=0} a(n)/(2*n)! = exp(1/3 + 2*sqrt(3)*Pi/27) = 2.08840341696864282...

Links

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

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.

Formula

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

where L(x) = x^2/(4-x^2) + 4*x*arctan(x/sqrt(4-x^2))/sqrt((4-x^2)^3)

from a formula given in the Sprugnoli link.

Example

E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 111*x^6/6! + 3089*x^8/8! + 131985*x^10/10! + 7977687*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
where
log(A(x)) = x^2/2 + x^4/6 + x^6/20 + x^8/70 + x^10/252 + x^12/924 + x^14/3432 + x^16/12870 +...+ x^(2*n)/A000984(n) +...
In closed form,
log(A(x)) = x^2/(4-x^2) + 4*x*arctan(x/sqrt(4-x^2))/sqrt((4-x^2)^3).

Prog

(PARI) {a(n)=(2*n)!*polcoeff(exp(sum(m=1, n, 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=x^2/(4-x^2 +Ox) + 4*x*atan(x/sqrt(4-x^2 +Ox))/sqrt((4-x^2 +Ox)^3)); (2*n)!*polcoeff(exp(L), 2*n)}

Crossrefs

Cf. A193442, A193443, A193444, A000984 (C(2*n,n)).

Sequence in context: A112463 A009471 A301990 * A355458 A260027 A061233

Adjacent sequences: A193438 A193439 A193440 * A193442 A193443 A193444

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2011

Status

approved