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A174346 Triangle T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

%I A174346 #9 Sep 08 2022 08:45:51
%S A174346 1,1,1,1,9,1,1,18,18,1,1,30,180,30,1,1,45,450,450,45,1,1,63,945,4725,
%T A174346 945,63,1,1,84,1764,13230,13230,1764,84,1,1,108,3024,31752,142884,
%U A174346 31752,3024,108,1,1,135,4860,68040,428652,428652,68040,4860,135,1
%N A174346 Triangle T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.
%H A174346 G. C. Greubel, <a href="/A174346/b174346.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A174346 T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ).
%F A174346 T(n, n-k) = T(n, k).
%e A174346 Triangle begins as:
%e A174346   1;
%e A174346   1,   1;
%e A174346   1,   9,    1;
%e A174346   1,  18,   18,     1;
%e A174346   1,  30,  180,    30,      1;
%e A174346   1,  45,  450,   450,     45,      1;
%e A174346   1,  63,  945,  4725,    945,     63,     1;
%e A174346   1,  84, 1764, 13230,  13230,   1764,    84,    1;
%e A174346   1, 108, 3024, 31752, 142884,  31752,  3024,  108,   1;
%e A174346   1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1;
%t A174346 T[n_,k_]:= (Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>k-1, 3^(k-1), 3^(n-k)];
%t A174346 Table[T[n,k], {n,12}, {k,n}]//Flatten
%o A174346 (Magma)
%o A174346 function T(n,k)
%o A174346   if Floor(n/2) gt k-1 then return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(k-1);
%o A174346   else return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(n-k);
%o A174346   end if; return T;
%o A174346 end function;
%o A174346 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 26 2021
%o A174346 (Sage)
%o A174346 def A174346(n,k): return (1/n)*binomial(n,k)*binomial(n,k-1)*( 3^(k-1) if ((n//2)>k-1) else 3^(n-k) )
%o A174346 flatten([[A174346(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Nov 26 2021
%Y A174346 Cf. A081582.
%K A174346 nonn,tabl
%O A174346 1,5
%A A174346 _Roger L. Bagula_, Mar 16 2010
%E A174346 Edited by _G. C. Greubel_, Nov 26 2021