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Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).
2

%I #6 Feb 08 2023 18:11:10

%S 1,0,2,0,4,3,0,12,18,4,0,48,108,48,5,0,240,720,480,100,6,0,1440,5400,

%T 4800,1500,180,7,0,10080,45360,50400,21000,3780,294,8,0,80640,423360,

%U 564480,294000,70560,8232,448,9,0,725760,4354560,6773760,4233600,1270080,197568,16128,648,10

%N Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

%C A refinement of the number of partial permutations of an n-set (A002720).

%C Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 0, 2;

%e [2] 0, 4, 3;

%e [3] 0, 12, 18, 4;

%e [4] 0, 48, 108, 48, 5;

%e [5] 0, 240, 720, 480, 100, 6;

%e [6] 0, 1440, 5400, 4800, 1500, 180, 7;

%e [7] 0, 10080, 45360, 50400, 21000, 3780, 294, 8;

%e [8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;

%p T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):

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

%Y Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.

%Y Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Feb 08 2023