A152663 Number of leading odd entries in all permutations of {1,2,...,n} (see example).

1, 1, 6, 16, 120, 540, 5040, 32256, 362880, 3024000, 39916800, 410572800, 6227020800, 76281004800, 1307674368000, 18598035456000, 355687428096000, 5762136335155200, 121645100408832000, 2211729098342400000, 51090942171709440000, 1030334000462807040000

Offset

1,3

Comments

a(n) = Sum_{k=0..ceiling(n/2)} k*A152662(n,k).

Links

Table of n, a(n) for n=1..22.

Formula

a(2n+1) = (2n+1)!;

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

D-finite with recurrence 2*(n+2)*a(n) +3*(-n-1)*a(n-1) -2*n*(n-1)*(n+1)*a(n-2) +(n-2)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2022

Example

a(3) = 6 because in the permutations 123, 132, 213, 231, 312, 321 we have 1+2+0+0+2+1 = 6 leading odd entries.

Maple

ao := proc (n) options operator, arrow; factorial(2*n+1) end proc: ae := proc (n) options operator, arrow: n*factorial(2*n)/(n+1) end proc: a := proc (n) if `mod`(n, 2) = 1 then ao((1/2)*n-1/2) else ae((1/2)*n) end if end proc: seq(a(n), n = 1 .. 20);

Mathematica

a[n_] := If[OddQ[n], n!, n*n!/(n+2)];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 02 2024 *)

Crossrefs

Cf. A152662, A152664, A152665.

Sequence in context: A239027 A218976 A173737 * A229560 A113561 A347923

Adjacent sequences: A152660 A152661 A152662 * A152664 A152665 A152666

Keyword

nonn

Author

Emeric Deutsch, Dec 13 2008

Status

approved