A174861 Triangle read by rows: T(n,k) = n! * (Stirling1(n,k) + Stirling1(n,n-k) - 1) + 1 with T(0,0)=1.

1, 1, 1, 1, -5, 1, 1, -11, -11, 1, 1, -311, 505, -311, 1, 1, 1561, -1919, -1919, 1561, 1, 1, -97919, 257761, -324719, 257761, -97919, 1, 1, 3517921, -8013599, 4475521, 4475521, -8013599, 3517921, 1, 1, -204382079, 539844481, -608549759, 545811841, -608549759, 539844481, -204382079, 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, -5, 1},
  {1, -11, -11, 1},
  {1, -311, 505, -311, 1},
  {1, 1561, -1919, -1919, 1561, 1},
  {1, -97919, 257761, -324719, 257761, -97919, 1},
  ...

Mathematica

t[n_, m_] = n!*(StirlingS1[n, m] + StirlingS1[n, n - m] - (StirlingS1[ n, 0] + StirlingS1[n, n]) + 1) - n! + 1;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A008275.

Sequence in context: A296974 A146954 A174949 * A110522 A146987 A297915

Adjacent sequences: A174858 A174859 A174860 * A174862 A174863 A174864

Keyword

sign,tabl,less

Author

Roger L. Bagula, Mar 31 2010

Extensions

Edited by Sean A. Irvine, Mar 15 2026

Status

approved