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A318391 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.

%I A318391 #12 Jan 22 2023 20:50:16
%S A318391 1,1,3,1,9,15,1,21,90,113,1,45,375,1130,1153,1,93,1350,7345,17295,
%T A318391 15125,1,189,4515,39550,161420,317625,245829,1,381,14490,192213,
%U A318391 1210650,4023250,6883212,4815403,1,765,45375,878010,8014503,40020750,113572998,173354508,111308699
%N A318391 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.
%H A318391 Andrew Howroyd, <a href="/A318391/b318391.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A318391 T(n,k) = S(n,k) * Sum_{i=1..k} s(k,i) * B(i)^2 where S = A008277, s = A048994, B = A000110.
%e A318391 The T(3,2) = 9 pairs of set partitions:
%e A318391   {{1},{2,3}}  {{1},{2,3}}
%e A318391   {{1},{2,3}}   {{1,2,3}}
%e A318391   {{1,2},{3}}  {{1,2},{3}}
%e A318391   {{1,2},{3}}   {{1,2,3}}
%e A318391   {{1,3},{2}}  {{1,3},{2}}
%e A318391   {{1,3},{2}}   {{1,2,3}}
%e A318391    {{1,2,3}}   {{1},{2,3}}
%e A318391    {{1,2,3}}   {{1,2},{3}}
%e A318391    {{1,2,3}}   {{1,3},{2}}
%e A318391 Triangle begins:
%e A318391      1
%e A318391      1     3
%e A318391      1     9    15
%e A318391      1    21    90   113
%e A318391      1    45   375  1130  1153
%e A318391      1    93  1350  7345 17295 15125
%t A318391 Table[StirlingS2[n,k]*Sum[StirlingS1[k,i]*BellB[i]^2,{i,k}],{n,10},{k,n}]
%o A318391 (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 A318391 { for(n=1, 10, print(row(n))) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A318391 Row sums are A001247. Last column is A059849.
%Y A318391 Cf. A000110, A000258, A008277, A048994, A060639, A181939, A318389, A318390, A318392, A318393.
%K A318391 nonn,tabl
%O A318391 1,3
%A A318391 _Gus Wiseman_, Aug 25 2018