A084417 Triangle read by rows: T(n,k) = Sum_{i=1..k} (n+1-i)! * Stirling2(n,n+1-i), n>=1, 1<=k<=n.

1, 2, 3, 6, 12, 13, 24, 60, 74, 75, 120, 360, 510, 540, 541, 720, 2520, 4080, 4620, 4682, 4683, 5040, 20160, 36960, 45360, 47166, 47292, 47293, 40320, 181440, 372960, 498960, 539784, 545580, 545834, 545835, 362880, 1814400, 4142880, 6048000, 6882120, 7068600, 7086750, 7087260, 7087261

Offset

1,2

Comments

Interpolates between A000670 and factorials.

Links

Table of n, a(n) for n=1..45.

Formula

T(n, k) = Sum_{i=1..k} (n+1-i)! * Stirling2(n, n+1-i), n>=1, 1<=k<=n.

abs(Sum_{k=1..n} (-1)^k*T(n,k)) = A089677(n). - Alois P. Heinz, Apr 12 2026

Example

Triangle begins:
    1;
    2,   3;
    6,  12,  13;
   24,  60,  74,  75;
  120, 360, 510, 540, 541;
  ...

Maple

T:=(n, k)->add((n+1-i)!*Stirling2(n, n+1-i), i=1..k):
seq(seq(T(n, k), k=1..n), n=1..10);

Crossrefs

Mirror image of array in A084416.

Cf. A000142 (first column), A000670 (main diagonal), A052875, A089677.

Row sums give A069321.

Sequence in context: A369899 A102779 A287109 * A324243 A015774 A015767

Adjacent sequences: A084414 A084415 A084416 * A084418 A084419 A084420

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jun 24 2003

Extensions

Edited by Emeric Deutsch, May 11 2004

Edited by Sean A. Irvine, Apr 12 2026

Status

approved