|
%I #9 Apr 17 2021 21:52:45
%S 1,1,1,1,2,1,1,6,6,1,1,12,36,12,1,1,20,120,120,20,1,1,30,300,600,300,
%T 30,1,1,42,630,2100,2100,630,42,1,1,56,1176,5880,9800,5880,1176,56,1,
%U 1,72,2016,14112,35280,35280,14112,2016,72,1,1,90,3240,30240,105840,158760,105840,30240,3240,90,1
%N Triangle T(n,k) = k*binomial(n,k)*binomial(n-1,k) with T(n,0) = T(n,n) = 1, read by rows.
%H G. C. Greubel, <a href="/A173881/b173881.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1) with c(0) = c(1) = 1.
%F From _G. C. Greubel_, Apr 17 2021: (Start)
%F T(n,k) = k*binomial(n,k)*binomial(n-1,k) with T(n,0) = T(n,n) = 1.
%F Sum_{k=0..n} T(n,k) = 2*(2*n-3)*binomial(2*n-4, n-2) + 2 - [n=0] = 2 + 2A002457(n-2) - [n=0]. (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 6, 6, 1;
%e 1, 12, 36, 12, 1;
%e 1, 20, 120, 120, 20, 1;
%e 1, 30, 300, 600, 300, 30, 1;
%e 1, 42, 630, 2100, 2100, 630, 42, 1;
%e 1, 56, 1176, 5880, 9800, 5880, 1176, 56, 1;
%e 1, 72, 2016, 14112, 35280, 35280, 14112, 2016, 72, 1;
%e 1, 90, 3240, 30240, 105840, 158760, 105840, 30240, 3240, 90, 1;
%e ...
%t T[n_, k_]:= If[k==0 || k==n, 1, k*Binomial[n, k]*Binomial[n-1, k]];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 17 2021 *)
%o (Magma)
%o T:= func< n,k | k eq 0 or k eq n select 1 else k*Binomial(n,k)*Binomial(n-1,k) >;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 17 2021
%o (Sage)
%o def T(n,k): return 1 if (k==0 or k==n) else k*binomial(n,k)*binomial(n-1,k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 17 2021
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 01 2010
%E Edited by _G. C. Greubel_, Apr 17 2021
|
