A173881 - OEIS

%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