A318392 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.

1, 3, 1, 15, 9, 1, 119, 87, 18, 1, 1343, 1045, 285, 30, 1, 19905, 15663, 4890, 705, 45, 1, 369113, 286419, 95613, 16450, 1470, 63, 1, 8285261, 6248679, 2147922, 410053, 44870, 2730, 84, 1, 219627683, 159648795, 55211229, 11202534, 1394883, 105714, 4662, 108, 1

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}}  {{1,3},{2}}
   {{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},{2},{3}}
   {{1,3},{2}}   {{1,3},{2}}
Triangle begins:
      1
      3     1
     15     9     1
    119    87    18     1
   1343  1045   285    30     1
  19905 15663  4890   705    45     1
		

Crossrefs

Row sums are A001247. First column is A060639.

Cf. A000110, A000258, A008277, A048994, A059849, A181939, A318389, A318390, A318391, A318393.

Programs

Mathematica

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}]

Formula

E.g.f.: (Sum_{n>=0} B(n)^2 x^n/n!)^t where B = A000110.