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

1, 1, 3, 1, 9, 15, 1, 21, 90, 113, 1, 45, 375, 1130, 1153, 1, 93, 1350, 7345, 17295, 15125, 1, 189, 4515, 39550, 161420, 317625, 245829, 1, 381, 14490, 192213, 1210650, 4023250, 6883212, 4815403, 1, 765, 45375, 878010, 8014503, 40020750, 113572998, 173354508, 111308699

Offset: 1

Author

Gus Wiseman, Aug 25 2018

Keywords

• nonn
• tabl

Examples

			The T(3,2) = 9 pairs of set partitions:
  {{1},{2,3}}  {{1},{2,3}}
  {{1},{2,3}}   {{1,2,3}}
  {{1,2},{3}}  {{1,2},{3}}
  {{1,2},{3}}   {{1,2,3}}
  {{1,3},{2}}  {{1,3},{2}}
  {{1,3},{2}}   {{1,2,3}}
   {{1,2,3}}   {{1},{2,3}}
   {{1,2,3}}   {{1,2},{3}}
   {{1,2,3}}   {{1,3},{2}}
Triangle begins:
     1
     1     3
     1     9    15
     1    21    90   113
     1    45   375  1130  1153
     1    93  1350  7345 17295 15125
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A001247. Last column is A059849.

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

Programs

Mathematica

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

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))}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = S(n,k) * Sum_{i=1..k} s(k,i) * B(i)^2 where S = A008277, s = A048994, B = A000110.