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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)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.
2

%I #33 Jul 18 2022 23:40:15

%S 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 1750,1680,672,128,7,236,1876,5838,8400,5824,1792,256,8,378,4056,

%U 17136,34524,35616,18816,4608,512,9,580,8190,45480,122682,175896,137760,57600,11520,1024

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

%H Jordan Weaver, <a href="/A353132/b353132.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) = A353131(n,k)/(n-k+1)!

%F Sum_{k=1..n} T(n,k) = A349458(n).

%e 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 Triangle begins:

%e [1] 2;

%e [2] 1, 4;

%e [3] 2, 6, 8;

%e [4] 3, 18, 24, 16;

%e [5] 4, 40, 100, 80, 32;

%e [6] 5, 78, 305, 440, 240, 64;

%e [7] 6, 140, 798, 1750, 1680, 672, 128;

%e [8] 7, 236, 1876, 5838, 8400, 5824, 1792, 256;

%e [9] 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512;

%e [10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.

%Y Cf. A000079, A353131, A349413, A268441, A178867.

%K nonn,tabl

%O 1,1

%A _Jordan Weaver_, Apr 24 2022