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URL: https://oeis.org/A193639

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A193639
Triangle T(n,k) of ways n couples can sit in a row with exactly k of them together.
2
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
0,3
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.
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).
EXAMPLE
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.
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 *)
CROSSREFS
Sequence in context: A289841 A143812 A245508 * A388370 A154481 A394547
KEYWORD
nonn,tabl
AUTHOR
Andrew Woods, Aug 01 2011
STATUS
approved