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A173882 Triangle T(n, k) = A090443(n-1)/(A090443(k-1)*A090443(n-k-1)) read by rows.

%I A173882 #14 Apr 17 2021 21:49:35
%S A173882 1,1,1,1,6,1,1,24,24,1,1,60,240,60,1,1,120,1200,1200,120,1,1,210,4200,
%T A173882 10500,4200,210,1,1,336,11760,58800,58800,11760,336,1,1,504,28224,
%U A173882 246960,493920,246960,28224,504,1,1,720,60480,846720,2963520,2963520,846720,60480,720,1
%N A173882 Triangle T(n, k) = A090443(n-1)/(A090443(k-1)*A090443(n-k-1)) read by rows.
%C A173882 A090443 is defined as +1 at negative indices here, which keeps the definition valid in the range 0 <= k <= n.
%C A173882 Row sums are 1, 2, 8, 50, 362, 2642, 19322, 141794, 1045298, 7742882, ....
%H A173882 G. C. Greubel, <a href="/A173882/b173882.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173882 T(n, k) = 2*binomial(n-1,k-1)*binomial(n,k)*binomial(n+1,k+1)*(n-k)/(n-k+1) with T(n, 0) = T(n, n) = 1.
%F A173882 T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} (j-1)*j*(j+1) = (n-1)!*n!*(n+1)!/2 and c(0) = c(1) = 1.
%e A173882 Triangle begins as:
%e A173882   1;
%e A173882   1,   1;
%e A173882   1,   6,      1;
%e A173882   1,  24,     24,       1;
%e A173882   1,  60,    240,      60,        1;
%e A173882   1, 120,   1200,    1200,      120,        1;
%e A173882   1, 210,   4200,   10500,     4200,      210,        1;
%e A173882   1, 336,  11760,   58800,    58800,    11760,      336,       1;
%e A173882   1, 504,  28224,  246960,   493920,   246960,    28224,     504,      1;
%e A173882   1, 720,  60480,  846720,  2963520,  2963520,   846720,   60480,    720,   1;
%e A173882   1, 990, 118800, 2494800, 13970880, 24449040, 13970880, 2494800, 118800, 990, 1;
%e A173882   ...
%p A173882 A090443 := proc(n) (n+2)!*(n+1)!*n!/2 ; end proc:
%p A173882 A173882 := proc(n,m) if m=0 or m= n then 1; else A090443(n-1)/A090443(m-1)/A090443(n-m-1) ; end if; end proc:
%p A173882 seq(seq(A173882(n,m),m=0..n),n=0..5) ; # _R. J. Mathar_, Mar 19 2011
%t A173882 T[n_,k_]:= If[k==0||k==n, 1, 2*Binomial[n-1,k-1]*Binomial[n,k]*Binomial[n+1,k+1]*(n-k)/(n-k+1)];
%t A173882 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 17 2021 *)
%o A173882 (Magma)
%o A173882 T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n-1,k-1)*Binomial(n,k)*Binomial(n+1,k+1)*(n-k)/(n-k+1) >;
%o A173882 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 17 2021
%o A173882 (Sage)
%o A173882 def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n-1,k-1)*binomial(n,k)*binomial(n+1,k+1)*(n-k)/(n-k+1)
%o A173882 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 17 2021
%Y A173882 Cf. A056939.
%K A173882 nonn,tabl,easy
%O A173882 0,5
%A A173882 _Roger L. Bagula_, Mar 01 2010