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A350266 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.

%I A350266 #13 Jan 09 2022 22:23:54
%S A350266 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 A350266 2160,720,0,7,147,1470,7350,17640,17640,5040,0,8,224,3136,23520,94080,
%U A350266 188160,161280,40320,0,9,324,6048,63504,381024,1270080,2177280,1632960,362880
%N A350266 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 A350266 T(n, k) = binomial(n, k)^2 * k! / (n - k + 1) if k >= 1.
%e A350266 Table starts:
%e A350266 [0] 1;
%e A350266 [1] 0, 1;
%e A350266 [2] 0, 2,   2;
%e A350266 [3] 0, 3,   9,    6;
%e A350266 [4] 0, 4,  24,   48,    24;
%e A350266 [5] 0, 5,  50,  200,   300,    120;
%e A350266 [6] 0, 6,  90,  600,  1800,   2160,     720;
%e A350266 [7] 0, 7, 147, 1470,  7350,  17640,   17640,    5040;
%e A350266 [8] 0, 8, 224, 3136, 23520,  94080,  188160,  161280,   40320;
%e A350266 [9] 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880;
%p A350266 T := (n, k) -> ifelse(k = 0, k^n, binomial(n, k)^2 * k! / (n - k + 1)):
%p A350266 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A350266 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 A350266 A350267 (row sums), A000142 (main diagonal), A074143 (subdiagonal), A006002 (column 2), A089835 (central terms).
%K A350266 nonn,tabl
%O A350266 0,5
%A A350266 _Peter Luschny_, Jan 09 2022