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A353131 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.

%I A353131 #30 Jul 19 2022 01:13:09
%S A353131 2,2,4,12,12,8,72,108,48,16,480,960,600,160,32,3600,9360,7320,2640,
%T A353131 480,64,30240,100800,95760,42000,10080,1344,128,282240,1189440,
%U A353131 1350720,700560,201600,34944,3584,256,2903040,15240960,20442240,12337920,4142880,854784,112896,9216,512
%N A353131 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 A353131 Jordan Weaver, <a href="/A353131/b353131.txt">Rows 1 to 40 of triangle, flattened</a>
%H A353131 E. T. Bell, <a href="https://www.jstor.org/stable/1967979">Partition polynomials</a>, Ann. Math., 29 (1927-1928), 38-46.
%H A353131 E. T. Bell, <a href="https://www.jstor.org/stable/1968431">Exponential polynomials</a>, Ann. Math., 35 (1934), 258-277.
%H A353131 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 A353131 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 A353131 A. Postnikov, <a href="https://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.
%F A353131 T(n,k) = A353132(n,k)*(n-k+1)!.
%F A353131 Sum_{k=1..n} T(n,k)/(n-k+1)! = A349458(n).
%e A353131 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 A353131 Triangle starts:
%e A353131 [1]      2;
%e A353131 [2]      2,       4;
%e A353131 [3]     12,      12,       8;
%e A353131 [4]     72,     108,      48,     16;
%e A353131 [5]    480,     960,     600,    160,     32;
%e A353131 [6]   3600,    9360,    7320,   2640,    480,    64;
%e A353131 [7]  30240,  100800,   95760,  42000,  10080,  1344,  128;
%e A353131 [8] 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256.
%Y A353131 Cf. A000079, A353132, A349413, A268441, A178867.
%K A353131 nonn,tabl
%O A353131 1,1
%A A353131 _Jordan Weaver_, Apr 24 2022