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

%I #11 Jan 27 2020 16:03:29

%S 1,1,5,51,857,21045,702597,30379839,1642718865,108171928521,

%T 8495805003525,782625366185355,83400601634195049,10163125433194019325,

%U 1402348965454767334725,217258436356989650347095,37513434482581646048138145,7172552434209226974773867025

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

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * k! * a(n-k). - _Ilya Gutkovskiy_, Jan 27 2020

%e E.g.f.: A(x) = 1 + x^2/2! + 5*x^4/4! + 51*x^6/6! + 857*x^8/8! + 21045*x^10/10! + 702597*x^12/12! +...+ a(n)*x^n/(2*n)! +...

%e where

%e log(A(x)) = x^2/2! + 2!*x^4/4! + 3!*x^6/6! + 4!*x^8/8! + 5!*x^10/10! +...

%o (PARI) {a(n)=(2*n)!*polcoeff(exp(sum(m=1, n, m!*x^(2*m)/(2*m)!)+O(x^(2*n+1))), 2*n)}

%Y Cf. A193441, A193442, A193443, A001813 ((2*n)!/n!).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 25 2011