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%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
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