A192990 Pyramid P(n, t, d) read by planes and rows, for 0 <= t+d <= n: number of ways n triples can sit in a row so that exactly t triples are together and exactly d triples are separated into a couple and a loner.

1, 0, 0, 6, 72, 144, 288, 0, 144, 72, 37584, 95904, 98496, 51840, 11664, 25920, 31104, 1296, 7776, 1296, 53529984, 127899648, 130761216, 69921792, 17915904, 11321856, 26002944, 23887872, 10202112, 1430784, 2985984, 2612736, 124416, 373248, 31104

Offset: 0

Author

Andrew Woods, Aug 02 2011

Keywords

• nonn
• tabf

Comments

The plane P(n,,) contains (n+1)*(n+2)/2 numbers.

The row P(n,t,) contains n+1-t numbers.

P(n,t,d) = a((n+1)*(n+2)*(n+3)/6 - (n-t+1)*(n-t+2)/2 + d)

The plane P(n,,) sums to (3n)!

Examples

			Pyramid starts:
1...0 0...72 144 288...37584 95904 98496 51840
....6..... 0 144.......11664 25920 31104
..........72........... 1296  7776
....................... 1296
There are P(3,1,2) = 31104 ways to arrange three sets of triples in a row so that one is together and two are split into a couple and a loner.
		

Links

• Andrew Woods, Table of n, a(n) for n = 0..1770, i.e. from P(0,,) to P(20,,)

Crossrefs

P(n,0,0) = A193624(n).