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%I #34 Feb 13 2024 08:14:04
%S 0,1,4,21,148,1333,14664,190633,2859496,48611433,923617228,
%T 19395961789,446107121148,11152678028701,301122306774928,
%U 8732546896472913,270708953790660304,8933395475091790033,312668841628212651156,11568747140243868092773
%N a(0) = 0; thereafter, a(n) = (2*n-1)*a(n-1) + 1.
%H Seiichi Manyama, <a href="/A286286/b286286.txt">Table of n, a(n) for n = 0..404</a>
%F a(n) = (2*n-1)!! * Sum_{k=1..n} 1/(2*k-1)!!. - _Seiichi Manyama_, Sep 02 2017
%F a(n) = floor((2*n-1)!!*A060196), for n > 0. - _Peter McNair_, Dec 10 2021
%F From _Peter Bala_, Feb 09 2024: (Start)
%F a(n) = 2*n*a(n-1) - (2*n - 3)*a(n-2) with a(0) = 0 and a(1) = 1.
%F The double factorial numbers (2*n-1)!! = A001147(n) satisfy the same recurrence, leading to the generalized continued fraction expansion Limit_{n -> oo} a(n)/(2*n-1)!! = Sum_{k >= 1} 1/(2*k-1)!! = A060196 = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). (End)
%t NestList[{(2 #2 - 1) #1 + 1, #2 + 1} & @@ # &, {0, 1}, 19][[All, 1]] (* _Michael De Vlieger_, Dec 10 2021 *)
%Y Conjectured to give indices of records in A132424.
%Y Cf. A001147, A002627 (similar sequence), A000522, A060196.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, May 15 2017
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