A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Author

Emeric Deutsch, Apr 06 2008

Keywords

• nonn
• tabl

Comments

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

Examples

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...
		

Crossrefs

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

Programs

Maple

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Mathematica

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

Formula

T(n,k) = 2*(n-k-1)*(n-2)!.

T(n,0) = 2(n-1)! = A052849(n-1).

T(n,1) = A052582(n-2).

T(n,2) = A052609(n-2).

T(n,3) = 12*A005990(n-3).

T(n,4) = 48*A061206(n-5).

T(n,n-2) = 2(n-2)! (A052849).

Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).