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Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.
2

%I #10 Jun 15 2022 11:52:24

%S 1,1,1,1,3,7,1,6,25,90,1,10,65,350,1701,1,15,140,1050,6951,42525,1,21,

%T 266,2646,22827,179487,1323652,1,28,462,5880,63987,627396,5715424,

%U 49329280,1,36,750,11880,159027,1899612,20912320,216627840,2141764053

%N Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.

%F T(n, k) = Stirling2(n + k, n).

%e Triangle T(n, k) begins:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 3, 7;

%e [3] 1, 6, 25, 90;

%e [4] 1, 10, 65, 350, 1701;

%e [5] 1, 15, 140, 1050, 6951, 42525;

%e [6] 1, 21, 266, 2646, 22827, 179487, 1323652;

%e [7] 1, 28, 462, 5880, 63987, 627396, 5715424, 49329280;

%e [8] 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053;

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

%p seq(seq(T(n, k), k = 0..n), n = 0..8);

%Y T(n,1) = A000217, T(n,n) = A007820, A354978 (row sums), A048993.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Jun 15 2022