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%I #24 May 24 2021 06:32:38
%S 1,1,7,105,2345,69405,2559667,113073961,5820788545,342176336073,
%T 22616620648895,1660292619682697,134029227728536985,
%U 11800452870718122325,1125324001129006580475,115551341953019187183225,12711056625162235880359425,1491325482312555276046069905
%N Number of cycle-up-down permutations of [2n] having n cycles.
%C For the definition of cycle-up-down permutations see A186366.
%H Alois P. Heinz, <a href="/A344445/b344445.txt">Table of n, a(n) for n = 0..338</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F a(n) = (2n)! * [x^(2n) y^n] 1/(1-sin(x))^y.
%F a(n) = A186366(2n,n).
%F a(n) ~ c * d^n * (n-1)!, where d = 7.3270710411718682766548233722838416956334898839746535623751... and c = 0.14278148012337362269164226210064788025688590260058738... - _Vaclav Kotesovec_, May 22 2021
%e a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).
%p b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p add(b(o-1+j, u-j), j=1..u))
%p end:
%p g:= proc(n) option remember; expand(`if`(n=0, 1,
%p add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
%p end:
%p a:= n-> coeff(g(2*n), x, n):
%p seq(a(n), n=0..18);
%t Join[{1}, Table[Sum[2^(2*n - 2*j + 1) * StirlingS1[2*j,n] * Sum[(-1)^k * k^(2*n) / ((j+k)!*(j-k)!), {k, 0, j}], {j, Floor[n/2], n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 22 2021 *)
%Y Cf. A007820, A186366, A344532.
%K nonn
%O 0,3
%A _Alois P. Heinz_, May 19 2021
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