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%I #25 Feb 25 2020 21:32:26
%S 1,1,1,2,3,1,6,9,6,1,22,31,28,10,1,92,123,126,69,15,1,426,549,586,418,
%T 145,21,1,2146,2695,2892,2425,1165,272,28,1,11624,14319,15262,14058,
%U 8551,2826,469,36,1
%N Triangle related to Bell numbers; T(n,k) read by rows, n>=0, 0<=k<=n: T(n,k) = k*T(n-1,k) + Sum(0<=j, T(n-1,k-1+j)); T(0,0)=1, T(0,k)=0 if k>0.
%C With offset 1 for k, T(n,k) is the number of indecomposable set partitions of [n+2] in which 1 is in the k-th block when the blocks are arranged in order of increasing largest entry. For example, T(2,2)=3 counts 2/134, 23/14, 3/124; see Link. - _David Callan_, Aug 30 2014
%H David Callan, <a href="/A086211/a086211.pdf">A combinatorial interpretation for this sequence</a>
%H Chunyan Yan, Zhicong Lin, <a href="https://arxiv.org/abs/1912.03674">Inversion sequences avoiding pairs of patterns</a>, arXiv:1912.03674 [math.CO], 2019.
%F Sum(k=0..n, A000110(k)*T(n-k,0)) = A000110(n+1).
%F Sum_{k=0..n} T(n, k) = A074664(n+2). - _Philippe Deléham_, May 10 2005
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 6, 9, 6, 1;
%e 22, 31, 28, 10, 1;
%e 92, 123, 126, 69, 15, 1;
%e 426, 549, 586, 418, 145, 21, 1;
%e 2146, 2695, 2892, 2425, 1165, 272, 28, 1;
%e 11624, 14319, 15262, 14058, 8551, 2826, 469, 36, 1 ;
%e ...
%Y Cf. A000110.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Aug 27 2003, Jun 16 2007
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