A193639 Triangle T(n,k) of ways n couples can sit in a row with exactly k of them together.

1, 0, 2, 8, 8, 8, 240, 288, 144, 48, 13824, 15744, 8064, 2304, 384, 1263360, 1401600, 710400, 211200, 38400, 3840, 168422400, 183582720, 92620800, 28108800, 5529600, 691200, 46080, 30865121280, 33223034880, 16717639680, 5148057600, 1061222400, 149022720, 13547520, 645120

Offset: 0

Author

Andrew Woods, Aug 01 2011

Keywords

• nonn
• tabl

Comments

Row n sums to (2*n)!.

Dot product of row n and (0,1,2,3,...n) is equal to (2*n)!, for n > 0.

Dot product of row n and (0,0,1,2,...n-1) is equal to T(n,0), for n > 0.

Examples

			Triangle begins:
  1
  0 2
  8 8 8
  240 288 144 48
  13824 15744 8064 2304 384
There are T(3, 2) = 144 ways to arrange three couples in a row so that exactly two of them are together.
		

Links

• Andrew Woods, Rows n = 0..50 of triangle, flattened

Programs

Mathematica

Table[Table[Sum[(-1)^k Binomial[n-i,k](2n-i-k)! 2^(k+i),{k,0,n-i}]*Binomial[n,i],{i,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, Apr 21 2014 *)

Formula

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

T(n, n) = 2^n * n! = (2*n)!!.

T(n, k) = Sum_{i=k..n} (-1)^(i-k) * 2^i * (2*n-i)! * binomial(n, i) * binomial(i, k).

T(n, 0) = A007060(n).

T(n, n) = A000165(n).