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Triangle T(n, k) = (binomial(n,2))! / (k! * abs(k+1 - binomial(n,2))!), read by rows.
2

%I #10 Jul 16 2023 16:59:30

%S 1,1,1,3,6,3,6,30,60,60,10,90,360,840,1260,15,210,1365,5460,15015,

%T 30030,21,420,3990,23940,101745,325584,813960,28,756,9828,81900,

%U 491400,2260440,8288280,24864840,36,1260,21420,235620,1884960,11686752,58433760,242082720,847289520

%N Triangle T(n, k) = (binomial(n,2))! / (k! * abs(k+1 - binomial(n,2))!), read by rows.

%H G. C. Greubel, <a href="/A123146/b123146.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = (binomial(n+1,2))! / (k! * abs(k+1 - binomial(n+1,2))!).

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 3, 6, 3;

%e 6, 30, 60, 60;

%e 10, 90, 360, 840, 1260;

%e 15, 210, 1365, 5460, 15015, 30030;

%e 21, 420, 3990, 23940, 101745, 325584, 813960;

%e 28, 756, 9828, 81900, 491400, 2260440, 8288280, 24864840;

%t T[n_, k_]:= (n*(n+1)/2)!/(k!*(Abs[k+1 -(n*(n+1)/2)])!);

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (Magma) [Factorial(Binomial(n+1,2))/(Factorial(k)*Factorial(Abs(k+1 - Binomial(n+1,2)))): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 16 2023

%o (SageMath)

%o def A123146(n,k): return factorial(binomial(n+1,2))/(factorial(k)*factorial(abs(k+1 - binomial(n+1,2))))

%o flatten([[A123146(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 16 2023

%K nonn,tabl,easy

%O 0,4

%A _Roger L. Bagula_, Oct 01 2006

%E Edited by _G. C. Greubel_, Jul 16 2023