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%I #14 May 19 2023 15:11:41
%S 1,1,4,60,2016,120960,11404800,1556755200,290594304000,71137485619200,
%T 22117290983424000,8515157028618240000,3977233344443842560000,
%U 2215887149047283712000000,1451849260055780288102400000,1105220249217462744317952000000
%N Number of permutations of [2n] in which the size of the last cycle is n and the cycles are ordered by increasing smallest elements.
%C For n>0, 2*a(n) corresponds to the number of random walk labelings of the n-barbell graph. - _Sela Fried_, May 19 2023
%H Alois P. Heinz, <a href="/A322450/b322450.txt">Table of n, a(n) for n = 0..250</a>
%H Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2305.09971">Further results on random walk labelings</a>, arXiv:2305.09971 [math.CO], 2023.
%F a(n) = (2n)!/(n*(n+1)) for n>0, a(0) = 1.
%F a(n) = A136125(2n,n).
%F From _Amiram Eldar_, Jan 19 2021: (Start)
%F Sum_{n>=1} 1/a(n) = e/4 + sinh(1)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = cos(1)/4 + 3*sin(1)/4. (End)
%p a:= n-> `if`(n=0, 1, (2*n)!/(n*(n+1))):
%p seq(a(n), n=0..18);
%t a[0] = 1; a[n_] := (2*n)!/(n*(n+1)); Array[a, 16, 0] (* _Amiram Eldar_, Jan 19 2021 *)
%Y Cf. A136125.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 08 2018
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