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%I #5 Sep 15 2018 15:43:52
%S 1,3,1,15,9,1,119,87,18,1,1343,1045,285,30,1,19905,15663,4890,705,45,
%T 1,369113,286419,95613,16450,1470,63,1,8285261,6248679,2147922,410053,
%U 44870,2730,84,1,219627683,159648795,55211229,11202534,1394883,105714,4662,108,1
%N Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.
%F E.g.f.: (Sum_{n>=0} B(n)^2 x^n/n!)^t where B = A000110.
%e The T(3,2) = 9 pairs of set partitions:
%e {{1},{2},{3}} {{1},{2,3}}
%e {{1},{2},{3}} {{1,2},{3}}
%e {{1},{2},{3}} {{1,3},{2}}
%e {{1},{2,3}} {{1},{2},{3}}
%e {{1},{2,3}} {{1},{2,3}}
%e {{1,2},{3}} {{1},{2},{3}}
%e {{1,2},{3}} {{1,2},{3}}
%e {{1,3},{2}} {{1},{2},{3}}
%e {{1,3},{2}} {{1,3},{2}}
%e Triangle begins:
%e 1
%e 3 1
%e 15 9 1
%e 119 87 18 1
%e 1343 1045 285 30 1
%e 19905 15663 4890 705 45 1
%t nn=5;Table[n!*SeriesCoefficient[Sum[BellB[n]^2*x^n/n!,{n,0,nn}]^t,{x,0,n},{t,0,k}],{n,nn},{k,n}]
%Y Row sums are A001247. First column is A060639.
%Y Cf. A000110, A000258, A008277, A048994, A059849, A181939, A318389, A318390, A318391, A318393.
%K nonn,tabl
%O 1,2
%A _Gus Wiseman_, Aug 25 2018
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