A174791 Triangle ready by rows: T(n,k) = n! * binomial(n,k) * (Eulerian(n+1,k) - 1) + 1.

1, 1, 1, 1, 13, 1, 1, 181, 181, 1, 1, 2401, 9361, 2401, 1, 1, 33601, 361201, 361201, 33601, 1, 1, 514081, 12852001, 34776001, 12852001, 514081, 1, 1, 8678881, 454265281, 2755015201, 2755015201, 454265281, 8678881, 1, 1, 161602561, 16490718721, 199223055361, 440827833601, 199223055361, 16490718721, 161602561, 1

Offset

0,5

Comments

Triangle is symmetric.

Links

Table of n, a(n) for n=0..44.

Example

Triangle begins:
  {1},
  {1, 1},
  {1, 13, 1},
  {1, 181, 181, 1},
  {1, 2401, 9361, 2401, 1},
  {1, 33601, 361201, 361201, 33601, 1},
  {1, 514081, 12852001, 34776001, 12852001, 514081, 1},
  ...

Maple

T:= (n, k)-> n!*binomial(n, k)*(combinat[eulerian1](n+1, k)-1)+1:
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Mar 13 2026

Mathematica

<< DiscreteMath`Combinatorica`
t[n_, m_] = (n!^2/(m!(n - m)!))*Eulerian[n + 1, m] - (n!^2/(m!(n - m)!)) + 1
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
Flatten[%]

Crossrefs

Cf. A173018.

Sequence in context: A172300 A022176 A188646 * A015132 A066036 A010231

Adjacent sequences: A174788 A174789 A174790 * A174792 A174793 A174794

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Mar 29 2010

Extensions

Edited by Sean A. Irvine, Mar 13 2026

Status

approved