A174588 Triangle read by rows: T(n,k) = S(n,k) - S(n,0) + 1 where S(n,k) = Sum_{j=0..n} (-1)^j * (k-j)^n * binomial(n, j)* Eulerian1(n+1, j) with S(0,k)=1.

1, 1, 1, 1, 7, 1, 1, 181, 181, 1, 1, 751, 1761, 751, 1, 1, -323399, -354899, -354899, -323399, 1, 1, -20873579, -27315455, -27944027, -27315455, -20873579, 1, 1, 1429118713, 1312010281, 1219622545, 1219622545, 1312010281, 1429118713, 1, 1

Offset

0,5

Comments

Triangle is symmetrical.

Links

Table of n, a(n) for n=0..36.

Example

Triangle begins:
  {1},
  {1, 1},
  {1, 7, 1},
  {1, 181, 181, 1},
  {1, 751, 1761, 751, 1},
  {1, -323399, -354899, -354899, -323399, 1},
  {1, -20873579, -27315455, -27944027, -27315455, -20873579, 1},
  ...

Mathematica

<< DiscreteMath`Combinatorica`
t[n_, m_] = If[n == 0, 1, Sum[Binomial[n, k]*Eulerian[n + 1, k]*(-1)^k*(m + 1 - k)^n, {k, 0, n}]];
Table[Table[t[n, m] - t[n, -1] + 1, {m, -1, n - 1}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A174587, A173018 (Eulerian1).

Sequence in context: A331899 A111830 A212943 * A366316 A199001 A350048

Adjacent sequences: A174585 A174586 A174587 * A174589 A174590 A174591

Keyword

sign,tabl,less

Author

Roger L. Bagula, Mar 23 2010

Extensions

Edited by Sean A. Irvine, Mar 05 2026

Status

approved