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Triangle read by rows: T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1.
1

%I #22 Oct 09 2018 05:41:33

%S 1,2,3,5,10,6,15,37,31,10,52,151,160,75,15,203,674,856,520,155,21,877,

%T 3263,4802,3556,1400,287,28,4140,17007,28337,24626,11991,3290,490,36,

%U 21147,94828,175896,174805,101031,34671,6972,786,45,115975,562595,1146931,1279240,853315,350889,88977,13620,1200,55

%N Triangle read by rows: T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1.

%H J. East, R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

%e Triangle begins:

%e 1

%e 2 3

%e 5 10 6

%e 15 37 31 10

%e 52 151 160 75 15

%e 203 674 856 520 155 21

%e 877 3263 4802 3556 1400 287 28

%e 4140 17007 28337 24626 11991 3290 490 36

%e ...

%t T[n_, k_] := Sum[Binomial[n, j] StirlingS2[j, k] BellB[n-j], {j, k, n}];

%t Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Oct 09 2018 *)

%Y Same as A049020 (which is the main entry for this triangle) except the present sequence has an extra 1 at the end of each row. - _R. J. Mathar_ and _N. J. A. Sloane_, May 17 2016

%K nonn,tabl

%O 1,2

%A _N. J. A. Sloane_, Jul 04 2014