A357367 - OEIS

%I #11 Dec 10 2023 09:23:20

%S 1,0,2,0,6,12,0,24,120,120,0,120,1080,2520,1680,0,720,10080,40320,

%T 60480,30240,0,5040,100800,604800,1512000,1663200,665280,0,40320,

%U 1088640,9072000,33264000,59875200,51891840,17297280

%N Triangle read by rows. T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.

%F T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.

%e Triangle T(n, k) start:

%e [0] 1;

%e [1] 0,     2;

%e [2] 0,     6,      12;

%e [3] 0,    24,     120,     120;

%e [4] 0,   120,    1080,    2520,     1680;

%e [5] 0,   720,   10080,   40320,    60480,    30240;

%e [6] 0,  5040,  100800,  604800,  1512000,  1663200,   665280;

%e [7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;

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

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

%o (SageMath)

%o def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)

%o def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)

%o         for m in range(k + 1)))

%o for n in range(8): print([T(n, k) for k in range(n+1)])

%Y Cf. A032037 (row sums), A271703.

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Sep 26 2022