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Triangle read by rows of partial Bell polynomials B_{n,k} (x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, n>=1, 1<=k<=n.
2

%I #30 Jul 19 2022 01:13:09

%S 2,2,4,12,12,8,72,108,48,16,480,960,600,160,32,3600,9360,7320,2640,

%T 480,64,30240,100800,95760,42000,10080,1344,128,282240,1189440,

%U 1350720,700560,201600,34944,3584,256,2903040,15240960,20442240,12337920,4142880,854784,112896,9216,512

%N Triangle read by rows of partial Bell polynomials B_{n,k} (x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, n>=1, 1<=k<=n.

%H Jordan Weaver, <a href="/A353131/b353131.txt">Rows 1 to 40 of triangle, flattened</a>

%H E. T. Bell, <a href="https://www.jstor.org/stable/1967979">Partition polynomials</a>, Ann. Math., 29 (1927-1928), 38-46.

%H E. T. Bell, <a href="https://www.jstor.org/stable/1968431">Exponential polynomials</a>, Ann. Math., 35 (1934), 258-277.

%H Sara C. Billey and Jordan E. Weaver, <a href="https://arxiv.org/abs/2207.06508">Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs</a>, arXiv:2207.06508 [math.CO], 2022.

%H A. Knutson, T. Lam and D. Speyer, <a href="http://dx.doi.org/10.1112/S0010437X13007240">Positroid varieties: juggling and geometry</a>, Compos. Math. 149 (2013), no. 10, 1710-1752.

%H A. Postnikov, <a href="https://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.

%F T(n,k) = A353132(n,k)*(n-k+1)!.

%F Sum_{k=1..n} T(n,k)/(n-k+1)! = A349458(n).

%e For n=4,k=2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3)=4x_1x_3+3x_2^2, so a(4,2)=B_{4,2}(2,2,12)=4*2*12+3*2^2=108.

%e Triangle starts:

%e [1] 2;

%e [2] 2, 4;

%e [3] 12, 12, 8;

%e [4] 72, 108, 48, 16;

%e [5] 480, 960, 600, 160, 32;

%e [6] 3600, 9360, 7320, 2640, 480, 64;

%e [7] 30240, 100800, 95760, 42000, 10080, 1344, 128;

%e [8] 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256.

%Y Cf. A000079, A353132, A349413, A268441, A178867.

%K nonn,tabl

%O 1,1

%A _Jordan Weaver_, Apr 24 2022