A349782 Triangle read by rows, T(n, k) = Sum_{j=0..k} |Stirling1(n, j)|.

1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 6, 17, 23, 24, 0, 24, 74, 109, 119, 120, 0, 120, 394, 619, 704, 719, 720, 0, 720, 2484, 4108, 4843, 5018, 5039, 5040, 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320, 0, 40320, 149904, 268028, 335312, 357761, 362297, 362843, 362879, 362880

Offset: 0

Author

Peter Luschny, Dec 02 2021

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of permutations of n objects that contain at most k cycles.

Examples

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,    2;
[3] 0, 2,    5,     6;
[4] 0, 6,    17,    23,    24;
[5] 0, 24,   74,    109,   119,   120;
[6] 0, 120,  394,   619,   704,   719,   720;
[7] 0, 720,  2484,  4108,  4843,  5018,  5039,  5040;
[8] 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320;
		

Crossrefs

Row sums: A121586, central terms: A349783.

Cf. A132393, A048994, A008275, A000142, A033312, A000774, A081046, A130534.

Programs

Maple

T := (n, k) -> add(abs(Stirling1(n,j)), j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_] := Sum[Abs[StirlingS1[n, j]], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 09 2021 *)

PARI

T(n, k) = sum(j=0, k, abs(stirling(n, j, 1))); \\ Michel Marcus, Dec 09 2021

Formula

T(n,k) = Sum_{j=0..k} A132393(n,j). - Alois P. Heinz, Dec 10 2021