A350266 - OEIS

%I #13 Jan 09 2022 22:23:54

%S 1,0,1,0,2,2,0,3,9,6,0,4,24,48,24,0,5,50,200,300,120,0,6,90,600,1800,

%T 2160,720,0,7,147,1470,7350,17640,17640,5040,0,8,224,3136,23520,94080,

%U 188160,161280,40320,0,9,324,6048,63504,381024,1270080,2177280,1632960,362880

%N Triangle read by rows. T(n, k) = binomial(n, k) * n! / (n - k + 1)! if k >= 1, if k = 0 then T(n, k) = k^n. T(n, k) for 0 <= k <= n.

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

%e Table starts:

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 2,   2;

%e [3] 0, 3,   9,    6;

%e [4] 0, 4,  24,   48,    24;

%e [5] 0, 5,  50,  200,   300,    120;

%e [6] 0, 6,  90,  600,  1800,   2160,     720;

%e [7] 0, 7, 147, 1470,  7350,  17640,   17640,    5040;

%e [8] 0, 8, 224, 3136, 23520,  94080,  188160,  161280,   40320;

%e [9] 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880;

%p T := (n, k) -> ifelse(k = 0, k^n, binomial(n, k)^2 * k! / (n - k + 1)):

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%t T[n_, 0] := Boole[n == 0]; T[n_, k_] := Binomial[n, k]^2 * k!/(n - k + 1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jan 09 2022 *)

%Y A350267 (row sums), A000142 (main diagonal), A074143 (subdiagonal), A006002 (column 2), A089835 (central terms).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Jan 09 2022