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A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

%I A360205 #6 Feb 08 2023 18:11:10
%S A360205 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 A360205 4800,1500,180,7,0,10080,45360,50400,21000,3780,294,8,0,80640,423360,
%U A360205 564480,294000,70560,8232,448,9,0,725760,4354560,6773760,4233600,1270080,197568,16128,648,10
%N A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).
%C A360205 A refinement of the number of partial permutations of an n-set (A002720).
%C A360205 Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).
%e A360205 Triangle T(n, k) starts:
%e A360205 [0] 1;
%e A360205 [1] 0,     2;
%e A360205 [2] 0,     4,      3;
%e A360205 [3] 0,    12,     18,      4;
%e A360205 [4] 0,    48,    108,     48,      5;
%e A360205 [5] 0,   240,    720,    480,    100,     6;
%e A360205 [6] 0,  1440,   5400,   4800,   1500,   180,    7;
%e A360205 [7] 0, 10080,  45360,  50400,  21000,  3780,  294,   8;
%e A360205 [8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;
%p A360205 T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):
%p A360205 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A360205 Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.
%Y A360205 Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).
%K A360205 nonn,tabl
%O A360205 0,3
%A A360205 _Peter Luschny_, Feb 08 2023