A262034 Number of permutations of [n] beginning with at least ceiling(n/2) ascents.

1, 0, 1, 1, 4, 5, 30, 42, 336, 504, 5040, 7920, 95040, 154440, 2162160, 3603600, 57657600, 98017920, 1764322560, 3047466240, 60949324800, 106661318400, 2346549004800, 4151586700800, 99638080819200, 177925144320000, 4626053752320000, 8326896754176000

Offset: 0

Author

Alois P. Heinz, Sep 08 2015

Keywords

• nonn

Examples

			a(4) = 4: 1234, 1243, 1342, 2341.
a(5) = 5: 12345, 12354, 12453, 13452, 23451.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..733

Crossrefs

Cf. A001761, A102693, A262033, A262035.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 0, 1$2][n+1],
      2*((n^2-1)*a(n-2)-a(n-1))/(n+3))
    end:
seq(a(n), n=0..30);

Mathematica

np=Rest[With[{nn=30},CoefficientList[Series[(Exp[x^2](x+1)-x^4/2+x^2+x+1)/ x^3,{x,0,nn}],x] Range[0,nn]!]//Quiet];Join[{1},np] (* Harvey P. Dale, May 18 2019 *)

Formula

E.g.f.: (exp(x^2)*(x+1)-(x^4/2+x^2+x+1))/x^3.

a(n) = 2*((n^2-1)*a(n-2)-a(n-1))/(n+3) for n>3, a(0)=a(2)=a(3)=1, a(1)=0.

a(n) = n!/(n/2+1)! if n even, a(n) = floor(C(n+1,(n+1)/2)/(n+3)*((n-1)/2)!) if n odd.

a(2n) = A262033(2n) = A001761(n).

a(2n+1) = A102693(n+1).

Sum_{n>=2} 1/a(n) = (39*exp(1/4)*sqrt(Pi)*erf(1/2) - 6)/16, where erf is the error function. - Amiram Eldar, Dec 04 2022