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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).
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%I #15 Mar 30 2021 12:03:00

%S 3,12,12,30,120,30,60,600,600,60,105,2100,5250,2100,105,168,5880,

%T 29400,29400,5880,168,252,14112,123480,246960,123480,14112,252,360,

%U 30240,423360,1481760,1481760,423360,30240,360,495,59400,1247400,6985440,12224520,6985440,1247400,59400,495

%N 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).

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

%C .............................................

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

%C .............../.................\...........

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

%C ...............\................./...........

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

%C .............................................

%C 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).

%H Seiichi Manyama, <a href="/A197208/b197208.txt">Rows n = 2..141, flattened</a>

%F 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)).

%F 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).

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

%F Row sums are A197209.

%e .n\k.|....1......2......3......4......5......6

%e = = = = = = = = = = = = = = = = = = = = = = = =

%e ..2..|....3...

%e ..3..|...12.....12

%e ..4..|...30....120.....30

%e ..5..|...60....600....600.....60

%e ..6..|..105...2100...5250...2100....105

%e ..7..|..168...5880..29400..29400...5880....168

%e ...

%e T(4,3) = sqrt(1*3*6*10*5*1) = sqrt(900) = 30

%e ..............1..............

%e ............1...1............

%e ..........1...2...1..........

%e ........1...3...3____1.......

%e .............../......\......

%e ......1...4...6...4....1.....

%e ...............\....../......

%e ...1...5...10...10___5.....1.

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

%K nonn,easy,tabl

%O 2,1

%A _Peter Bala_, Oct 12 2011