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

1,3

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Row sums are A001247. Last column is A059849.

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

Sequence in context: A232598 A174510 A141237 * A157399 A288852 A162749

Adjacent sequences: A318388 A318389 A318390 * A318392 A318393 A318394

Keyword

nonn,tabl

Author

Gus Wiseman, Aug 25 2018

Status

approved