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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).

%I A197208 #15 Mar 30 2021 12:03:00
%S A197208 3,12,12,30,120,30,60,600,600,60,105,2100,5250,2100,105,168,5880,
%T A197208 29400,29400,5880,168,252,14112,123480,246960,123480,14112,252,360,
%U A197208 30240,423360,1481760,1481760,423360,30240,360,495,59400,1247400,6985440,12224520,6985440,1247400,59400,495
%N 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).
%C A197208 In Pascal's triangle, the product of the six entries surrounding C(n,k) is a perfect square.
%C A197208 .............................................
%C A197208 ..............C(n-1,k-1)____C(n-1,k).........
%C A197208 .............../.................\...........
%C A197208 ............C(n,k-1)...C(n,k)....C(n,k+1)....
%C A197208 ...............\................./...........
%C A197208 ..............C(n+1,k)______C(n+1,k+1).......
%C A197208 .............................................
%C A197208 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 A197208 Seiichi Manyama, <a href="/A197208/b197208.txt">Rows n = 2..141, flattened</a>
%F A197208 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 A197208 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 A197208 T(n,k) = 1/2*(n^3-n)*A056939(n-2,k-1), for n >= 2 and 1 <= k <= n-1.
%F A197208 Row sums are A197209.
%e A197208 .n\k.|....1......2......3......4......5......6
%e A197208 = = = = = = = = = = = = = = = = = = = = = = = =
%e A197208 ..2..|....3...
%e A197208 ..3..|...12.....12
%e A197208 ..4..|...30....120.....30
%e A197208 ..5..|...60....600....600.....60
%e A197208 ..6..|..105...2100...5250...2100....105
%e A197208 ..7..|..168...5880..29400..29400...5880....168
%e A197208 ...
%e A197208 T(4,3) = sqrt(1*3*6*10*5*1) = sqrt(900) = 30
%e A197208 ..............1..............
%e A197208 ............1...1............
%e A197208 ..........1...2...1..........
%e A197208 ........1...3...3____1.......
%e A197208 .............../......\......
%e A197208 ......1...4...6...4....1.....
%e A197208 ...............\....../......
%e A197208 ...1...5...10...10___5.....1.
%Y A197208 Cf. A007318, A056939, A197209 (row sums).
%K A197208 nonn,easy,tabl
%O A197208 2,1
%A A197208 _Peter Bala_, Oct 12 2011