login
A318391
Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.
12
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.
Sequence in context: A232598 A174510 A141237 * A157399 A288852 A162749
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 25 2018
STATUS
approved