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Triangle read by rows: T(n,k) (1 <= k <= n) is the number of 3-extensions of an n-set over all choices of 3-partitions of the n-set.
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%I #22 Oct 27 2021 11:24:58

%S 1,2,1,5,4,1,14,13,6,1,41,40,25,9,1,122,121,90,48,12,1,365,364,301,

%T 202,78,14,1,1094,1093,966,747,380,106,16,1,3281,3280,3025,2559,1571,

%U 592,141,18,1,9842,9841,9330,8362,5864,2755,906,180,20,1

%N Triangle read by rows: T(n,k) (1 <= k <= n) is the number of 3-extensions of an n-set over all choices of 3-partitions of the n-set.

%C See Lindquist et al. 1981 for precise definition.

%H Norman Lindquist and Gerard Sierksma, <a href="https://doi.org/10.1016/0097-3165(81)90015-7">Extensions of set partitions</a>, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table III.

%e Triangle begins:

%e 1,

%e 2,1,

%e 5,4,1,

%e 14,13,6,1,

%e 41,40,25,9,1,

%e 122,121,90,48,12,1,

%e 365,364,301,202,78,14,1,

%e 1094,1093,966,747,380,106,16,1,

%e 3281,3280,3025,2559,1571,592,141,18,1,

%e 9842,9841,9330,8362,5864,2755,906,180,20,1,

%e ...

%Y Cf. A055248.

%Y Column 1 = A007051, column 3 = A000392.

%K nonn,tabl

%O 1,2

%A _N. J. A. Sloane_, Oct 26 2021