A086211 - OEIS

A086211

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.

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, 145, 21, 1, 2146, 2695, 2892, 2425, 1165, 272, 28, 1, 11624, 14319, 15262, 14058, 8551, 2826, 469, 36, 1

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OFFSET

0,4

COMMENTS

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

LINKS

Table of n, a(n) for n=0..44.

David Callan, A combinatorial interpretation for this sequence

Chunyan Yan, Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

FORMULA

Sum(k=0..n, A000110(k)*T(n-k,0)) = A000110(n+1).

Sum_{k=0..n} T(n, k) = A074664(n+2). - Philippe Deléham, May 10 2005

EXAMPLE

Triangle begins:

1, 1;

2, 3, 1;

6, 9, 6, 1;