A361893 - OEIS

%I #10 Jul 30 2023 17:45:42

%S 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 3600,3600,720,0,7,252,3150,16800,37800,30240,5040,0,8,392,7056,58800,

%U 235200,423360,282240,40320,0,9,576,14112,169344,1058400,3386880,5080320,2903040,362880

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

%F T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).

%F T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).

%F T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

%e Triangle T(n, k) starts:

%e   [0] 1;

%e   [1] 0, 1;

%e   [2] 0, 2,   2;

%e   [3] 0, 3,  12,     6;

%e   [4] 0, 4,  36,    72,     24;

%e   [5] 0, 5,  80,   360,    480,     120;

%e   [6] 0, 6, 150,  1200,   3600,    3600,     720;

%e   [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;

%e   [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;

%e   [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

%p A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:

%p seq(seq(A361893(n,k), k = 0..n), n = 0..9);

%p # Using the egf.:

%p egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):

%p poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):

%p row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

%Y Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

%Y Cf. A144084, A021010.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Mar 28 2023