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A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

%I A361893 #10 Jul 30 2023 17:45:42
%S A361893 1,0,1,0,2,2,0,3,12,6,0,4,36,72,24,0,5,80,360,480,120,0,6,150,1200,
%T A361893 3600,3600,720,0,7,252,3150,16800,37800,30240,5040,0,8,392,7056,58800,
%U A361893 235200,423360,282240,40320,0,9,576,14112,169344,1058400,3386880,5080320,2903040,362880
%N A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.
%F A361893 T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
%F A361893 T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
%F A361893 T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).
%e A361893 Triangle T(n, k) starts:
%e A361893   [0] 1;
%e A361893   [1] 0, 1;
%e A361893   [2] 0, 2,   2;
%e A361893   [3] 0, 3,  12,     6;
%e A361893   [4] 0, 4,  36,    72,     24;
%e A361893   [5] 0, 5,  80,   360,    480,     120;
%e A361893   [6] 0, 6, 150,  1200,   3600,    3600,     720;
%e A361893   [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
%e A361893   [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
%e A361893   [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;
%p A361893 A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
%p A361893 seq(seq(A361893(n,k), k = 0..n), n = 0..9);
%p A361893 # Using the egf.:
%p A361893 egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
%p A361893 poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
%p A361893 row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);
%Y A361893 Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.
%Y A361893 Cf. A144084, A021010.
%K A361893 nonn,tabl
%O A361893 0,5
%A A361893 _Peter Luschny_, Mar 28 2023