A262033 Number of permutations of [n] beginning with at least floor(n/2) ascents.

1, 1, 1, 3, 4, 20, 30, 210, 336, 3024, 5040, 55440, 95040, 1235520, 2162160, 32432400, 57657600, 980179200, 1764322560, 33522128640, 60949324800, 1279935820800, 2346549004800, 53970627110400, 99638080819200, 2490952020480000, 4626053752320000

Offset: 0

Author

Alois P. Heinz, Sep 08 2015

Keywords

• nonn

Examples

			a(4) = 4: 1234, 1243, 1342, 2341.
a(5) = 20: 12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 23415, 23451, 23514, 23541, 24513, 24531, 34512, 34521.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..732

Crossrefs

Cf. A001761, A006963, A262034, A262035.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1,
      2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := n!/Ceiling[(n + 1)/2]!; Array[a, 30, 0] (* Amiram Eldar, Dec 04 2022 *)

Formula

E.g.f.: (x+1)*(exp(x^2)-1)/x^2.

a(n) = 2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)) for n>1, a(0)=a(1)=1.

a(n) = n!/ceiling((n+1)/2)!.

a(2n) = A262034(2n) = A001761(n).

a(2n+1) = A006963(n+2).

Sum_{n>=0} 1/a(n) = 7/4 + 13*exp(1/4)*sqrt(Pi)*erf(1/2)/8, where erf is the error function. - Amiram Eldar, Dec 04 2022