A197208 Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).

3, 12, 12, 30, 120, 30, 60, 600, 600, 60, 105, 2100, 5250, 2100, 105, 168, 5880, 29400, 29400, 5880, 168, 252, 14112, 123480, 246960, 123480, 14112, 252, 360, 30240, 423360, 1481760, 1481760, 423360, 30240, 360, 495, 59400, 1247400, 6985440, 12224520, 6985440, 1247400, 59400, 495

Offset: 2

Author

Peter Bala, Oct 12 2011

Keywords

• nonn
• easy
• tabl

Comments

In Pascal's triangle, the product of the six entries surrounding C(n,k) is a perfect square.

.............................................

..............C(n-1,k-1)____C(n-1,k).........

.............../.................\...........

............C(n,k-1)...C(n,k)....C(n,k+1)....

...............\................./...........

..............C(n+1,k)______C(n+1,k+1).......

.............................................

In fact, C(n-1,k-1)*C(n,k+1)*C(n+1,k) = C(n-1,k)*C(n+1,k+1)*C(n,k-1).

Examples

			.n\k.|....1......2......3......4......5......6
= = = = = = = = = = = = = = = = = = = = = = = =
..2..|....3...
..3..|...12.....12
..4..|...30....120.....30
..5..|...60....600....600.....60
..6..|..105...2100...5250...2100....105
..7..|..168...5880..29400..29400...5880....168
...
T(4,3) = sqrt(1*3*6*10*5*1) = sqrt(900) = 30
..............1..............
............1...1............
..........1...2...1..........
........1...3...3____1.......
.............../......\......
......1...4...6...4....1.....
...............\....../......
...1...5...10...10___5.....1.
		

Links

• Seiichi Manyama, Rows n = 2..141, flattened

Crossrefs

Cf. A007318, A056939, A197209 (row sums).

Formula

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

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

T(n,k) = 1/2*(n^3-n)*A056939(n-2,k-1), for n >= 2 and 1 <= k <= n-1.

Row sums are A197209.