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A353132 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)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.

%I A353132 #33 Jul 18 2022 23:40:15
%S A353132 2,1,4,2,6,8,3,18,24,16,4,40,100,80,32,5,78,305,440,240,64,6,140,798,
%T A353132 1750,1680,672,128,7,236,1876,5838,8400,5824,1792,256,8,378,4056,
%U A353132 17136,34524,35616,18816,4608,512,9,580,8190,45480,122682,175896,137760,57600,11520,1024
%N A353132 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)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.
%H A353132 Jordan Weaver, <a href="/A353132/b353132.txt">Rows 1 to 40 of triangle, flattened</a>
%H A353132 E. T. Bell, <a href="https://www.jstor.org/stable/1967979">Partition polynomials</a>, Ann. Math., 29 (1927-1928), 38-46.
%H A353132 E. T. Bell, <a href="https://www.jstor.org/stable/1968431">Exponential polynomials</a>, Ann. Math., 35 (1934), 258-277.
%H A353132 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 A353132 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 A353132 A. Postnikov, <a href="https://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.
%F A353132 T(n,k) = A353131(n,k)/(n-k+1)!
%F A353132 Sum_{k=1..n} T(n,k) = A349458(n).
%e A353132 For n = 4, k = 2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3) = 4*x_1*x_3 + 3*x_2^2, so T(4,2) = B_{4,2}(2,2,12) - (4*2*12 + 3*2^2)/3! = 18.
%e A353132 Triangle begins:
%e A353132    [1] 2;
%e A353132    [2] 1,   4;
%e A353132    [3] 2,   6,    8;
%e A353132    [4] 3,  18,   24,    16;
%e A353132    [5] 4,  40,  100,    80,     32;
%e A353132    [6] 5,  78,  305,   440,    240,     64;
%e A353132    [7] 6, 140,  798,  1750,   1680,    672,    128;
%e A353132    [8] 7, 236, 1876,  5838,   8400,   5824,   1792,   256;
%e A353132    [9] 8, 378, 4056, 17136,  34524,  35616,  18816,  4608,   512;
%e A353132   [10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.
%Y A353132 Cf. A000079, A353131, A349413, A268441, A178867.
%K A353132 nonn,tabl
%O A353132 1,1
%A A353132 _Jordan Weaver_, Apr 24 2022