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%I #13 Jul 27 2019 16:00:19
%S 1,6,1,54,18,1,648,324,36,1,9720,6480,1080,60,1,174960,145800,32400,
%T 2700,90,1,3674160,3674160,1020600,113400,5670,126,1,88179840,
%U 102876480,34292160,4762800,317520,10584,168,1,2380855680,3174474240,1234517760,205752960,17146080,762048,18144,216,1
%N The third 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, -2 <= d <= 3).
%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/(1-3*x)).
%F T(n,k) = 3^(n-k)*binomial(n-1, k-1)*n!/k! = 3^(n-k)*A105278.
%e Triangle begins:
%e 1;
%e 6, 1;
%e 54, 18, 1;
%e 648, 324, 36, 1;
%e 9720, 6480, 1080, 60, 1;
%e ...
%t Table[3^(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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