A362369 - OEIS

%I #11 May 10 2023 08:24:01

%S 1,0,0,2,0,3,0,4,12,0,5,60,0,6,210,120,0,7,630,1260,0,8,1736,8400,

%T 1680,0,9,4536,45360,30240,0,10,11430,216720,327600,30240,0,11,28050,

%U 956340,2772000,831600,0,12,67452,3993000,20207880,13305600,665280

%N Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

%F From _Mélika Tebni_, May 10 2023: (Start)

%F E.g.f. of column k: (x*(exp(x)-1))^k / k!.

%F Sum_{k=0..n-1} (-1)^(n+k-1)*T(n+k-1, k) = A000169(n), for n > 0. (End)

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 0;

%e [2] 0, 2;

%e [3] 0, 3;

%e [4] 0, 4,   12;

%e [5] 0, 5,   60;

%e [6] 0, 6,  210,   120;

%e [7] 0, 7,  630,  1260;

%e [8] 0, 8, 1736,  8400,  1680;

%e [9] 0, 9, 4536, 45360, 30240;

%p T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):

%p seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);

%p # second program:

%p egf := k-> (x*(exp(x)-1))^k / k!:

%p A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):

%p seq(print(seq(A362369(n, k), k=0..iquo(n,2))), n=0..12); # _Mélika Tebni_, May 10 2023

%o (SageMath)

%o def A362369(n, k):

%o     return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)

%o for n in range(10):

%o     print([A362369(n, k) for k in range(n//2 + 1)])

%Y Cf. A000169, A052506 (row sums), A362788, A362789.

%K nonn,tabf

%O 0,4

%A _Peter Luschny_, May 04 2023