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A174298 Triangle T(n, k) = binomial(n, k)*( n!/k! if floor(n/2) >= k otherwise n!/(n-k)! ), read by rows.

%I A174298 #7 Sep 08 2022 08:45:51
%S A174298 1,1,1,2,4,2,6,18,18,6,24,96,72,96,24,120,600,600,600,600,120,720,
%T A174298 4320,5400,2400,5400,4320,720,5040,35280,52920,29400,29400,52920,
%U A174298 35280,5040,40320,322560,564480,376320,117600,376320,564480,322560,40320,362880,3265920,6531840,5080320,1905120,1905120,5080320,6531840,3265920,362880
%N A174298 Triangle T(n, k) = binomial(n, k)*( n!/k! if floor(n/2) >= k otherwise n!/(n-k)! ), read by rows.
%H A174298 G. C. Greubel, <a href="/A174298/b174298.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A174298 T(n, k) = binomial(n, k)*( n!/k! if floor(n/2) >= k otherwise n!/(n-k)! ).
%F A174298 From _G. C. Greubel_, Nov 24 2021: (Start)
%F A174298 T(n, k) = binomial(n, k)^2*( (n-k)! if floor(n/2) >= k otherwise k! ).
%F A174298 T(n, 0) = T(n, n) = n!.
%F A174298 T(n, k) = T(n, n-k).
%F A174298 T(2*n, n) = (-1)^n*A295383(n). (End)
%e A174298 Triangle begins as:
%e A174298       1;
%e A174298       1,      1;
%e A174298       2,      4,      2;
%e A174298       6,     18,     18,      6;
%e A174298      24,     96,     72,     96,     24;
%e A174298     120,    600,    600,    600,    600,    120;
%e A174298     720,   4320,   5400,   2400,   5400,   4320,    720;
%e A174298    5040,  35280,  52920,  29400,  29400,  52920,  35280,   5040;
%e A174298   40320, 322560, 564480, 376320, 117600, 376320, 564480, 322560, 40320;
%t A174298 T[n_, k_]:= Binomial[n, k]*If[Floor[n/2]>=k, n!/k!, n!/(n-k)!];
%t A174298 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%o A174298 (Magma)
%o A174298 A174298:= func< n,k | Floor(n/2) gt k select Factorial(n-k)*Binomial(n,k)^2 else Factorial(k)*Binomial(n,k)^2 >;
%o A174298 [A174298(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 24 2021
%o A174298 (Sage)
%o A174298 def A174298(n,k): return binomial(n,k)^2*( factorial(n-k) if ((n//2) > k-1) else factorial(k))
%o A174298 flatten([[A174298(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 24 2021
%Y A174298 Cf. A295383.
%K A174298 nonn,tabl
%O A174298 0,4
%A A174298 _Roger L. Bagula_, Mar 15 2010
%E A174298 Edited by _G. C. Greubel_, Nov 24 2021