A308282 - OEIS

%I #13 Jul 27 2019 16:01:51

%S 1,10,1,150,30,1,3000,900,60,1,75000,30000,3000,100,1,2250000,1125000,

%T 150000,7500,150,1,78750000,47250000,7875000,525000,15750,210,1,

%U 3150000000,2205000000,441000000,36750000,1470000,29400,280,1,141750000000,113400000000,26460000000,2646000000,132300000,3528000,50400,360,1

%N The fifth power of the unsigned Lah triangular matrix A105278.

%C Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -4 <= d <= 5).

%H N. Nakashima and S. Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.

%F E.g.f.: exp(x*y/(1-5*x)).

%F T(n,k) = 5^(n-k)*binomial(n-1, k-1)*n!/k! = 5^(n-k)*A105278.

%e Triangle begins:

%e       1;

%e      10,     1;

%e     150,    30,    1;

%e    3000,   900,   60,   1;

%e   75000, 30000, 3000, 100, 1;

%e   ...

%t Table[5^(n - k) * Binomial[n - 1, k - 1] * n! / k!, {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 13 2019 *)

%Y Cf. A105278.

%K nonn,tabl,easy

%O 1,2

%A _Shuhei Tsujie_, May 18 2019