A344445 Number of cycle-up-down permutations of [2n] having n cycles.

1, 1, 7, 105, 2345, 69405, 2559667, 113073961, 5820788545, 342176336073, 22616620648895, 1660292619682697, 134029227728536985, 11800452870718122325, 1125324001129006580475, 115551341953019187183225, 12711056625162235880359425, 1491325482312555276046069905

Offset: 0

Author

Alois P. Heinz, May 19 2021

Keywords

• nonn

Comments

For the definition of cycle-up-down permutations see A186366.

Examples

			a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..338
• Wikipedia, Permutation

Crossrefs

Cf. A007820, A186366, A344532.

Programs

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
    end:
a:= n-> coeff(g(2*n), x, n):
seq(a(n), n=0..18);

Mathematica

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 *)

Formula

a(n) = (2n)! * [x^(2n) y^n] 1/(1-sin(x))^y.

a(n) = A186366(2n,n).

a(n) ~ c * d^n * (n-1)!, where d = 7.3270710411718682766548233722838416956334898839746535623751... and c = 0.14278148012337362269164226210064788025688590260058738... - Vaclav Kotesovec, May 22 2021