%I #12 Jan 22 2023 20:50:16

%S 1,1,3,1,9,15,1,21,90,113,1,45,375,1130,1153,1,93,1350,7345,17295,

%T 15125,1,189,4515,39550,161420,317625,245829,1,381,14490,192213,

%U 1210650,4023250,6883212,4815403,1,765,45375,878010,8014503,40020750,113572998,173354508,111308699

%N Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.

%H Andrew Howroyd, <a href="/A318391/b318391.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F T(n,k) = S(n,k) * Sum_{i=1..k} s(k,i) * B(i)^2 where S = A008277, s = A048994, 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}} {{1,2},{3}}

%e {{1,2},{3}} {{1,2,3}}

%e {{1,3},{2}} {{1,3},{2}}

%e {{1,3},{2}} {{1,2,3}}

%e {{1,2,3}} {{1},{2,3}}

%e {{1,2,3}} {{1,2},{3}}

%e {{1,2,3}} {{1,3},{2}}

%e Triangle begins:

%e 1

%e 1 3

%e 1 9 15

%e 1 21 90 113

%e 1 45 375 1130 1153

%e 1 93 1350 7345 17295 15125

%t Table[StirlingS2[n,k]*Sum[StirlingS1[k,i]*BellB[i]^2,{i,k}],{n,10},{k,n}]

%o (PARI) row(n) = {my(b=Vec(serlaplace(exp(exp(x + O(x*x^n))-1)-1))); vector(n, k, stirling(n,k,2)*sum(i=1, k, stirling(k,i,1)*b[i]^2))}

%o { for(n=1, 10, print(row(n))) } \\ _Andrew Howroyd_, Jan 19 2023

%Y Row sums are A001247. Last column is A059849.

%Y Cf. A000110, A000258, A008277, A048994, A060639, A181939, A318389, A318390, A318392, A318393.

%K nonn,tabl

%O 1,3

%A _Gus Wiseman_, Aug 25 2018