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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row n sums to (2n)!

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

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

LINKS

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

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! = (2n)!!

T(n, k) = sum(i=k..n, (-1)^(i-k) * 2^i * (2n-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 * A154481 A092280 A070987

Adjacent sequences:  A193636 A193637 A193638 * A193640 A193641 A193642

KEYWORD

nonn,tabl

AUTHOR

Andrew Woods, Aug 01 2011

STATUS

approved

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)