A174639 Triangle read by rows: T(n,k) = binomial(n,k) * bell(k+1) * bell(n-k+1) - bell(n+1) + 1 where bell(n) = A000110(n) are the Bell numbers.

1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 69, 99, 69, 1, 1, 318, 548, 548, 318, 1, 1, 1560, 3024, 3624, 3024, 1560, 1, 1, 8139, 17176, 23161, 23161, 17176, 8139, 1, 1, 45094, 101634, 149374, 168134, 149374, 101634, 45094, 1, 1, 264672, 629226, 989046, 1214082, 1214082, 989046, 629226, 264672, 1

Offset

0,5

Comments

Triangle is symmetric.

Links

Table of n, a(n) for n=0..54.

Example

Triangle begins:
  {1},
  {1, 1},
  {1, 4, 1},
  {1, 16, 16, 1},
  {1, 69, 99, 69, 1},
  {1, 318, 548, 548, 318, 1},
  {1, 1560, 3024, 3624, 3024, 1560, 1},
  {1, 8139, 17176, 23161, 23161, 17176, 8139, 1},
  ...

Maple

B:= proc(n) option remember; `if`(n=0, 1,
      add(B(n-j)*binomial(n-1, j-1), j=1..n))
    end:
T:= (n, k)-> binomial(n, k)*B(k+1)*B(n-k+1)-B(n+1)+1:
seq(seq(T(n, k), k=0..n), n=0..9);  # Alois P. Heinz, Mar 07 2026

Mathematica

f[n_] := Sum[StirlingS2[n, k], {k, 1, n}];
t[n_, m_] = Binomial[n, m]*f[m + 1]*f[n - m + 1]
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A000110.

Sequence in context: A203846 A118185 A176483 * A173814 A176467 A034802

Adjacent sequences: A174636 A174637 A174638 * A174640 A174641 A174642

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Mar 25 2010

Extensions

Edited by Sean A. Irvine, Mar 07 2026

Status

approved