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

%I A349782 #18 Dec 10 2021 18:54:06
%S A349782 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,
%T A349782 704,719,720,0,720,2484,4108,4843,5018,5039,5040,0,5040,18108,31240,
%U A349782 38009,39969,40291,40319,40320,0,40320,149904,268028,335312,357761,362297,362843,362879,362880
%N A349782 Triangle read by rows, T(n, k) = Sum_{j=0..k} |Stirling1(n, j)|.
%C A349782 T(n, k) is the number of permutations of n objects that contain at most k cycles.
%F A349782 T(n,k) = Sum_{j=0..k} A132393(n,j). - _Alois P. Heinz_, Dec 10 2021
%e A349782 Triangle starts:
%e A349782 [0] 1;
%e A349782 [1] 0, 1;
%e A349782 [2] 0, 1,    2;
%e A349782 [3] 0, 2,    5,     6;
%e A349782 [4] 0, 6,    17,    23,    24;
%e A349782 [5] 0, 24,   74,    109,   119,   120;
%e A349782 [6] 0, 120,  394,   619,   704,   719,   720;
%e A349782 [7] 0, 720,  2484,  4108,  4843,  5018,  5039,  5040;
%e A349782 [8] 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320;
%p A349782 T := (n, k) -> add(abs(Stirling1(n,j)), j = 0..k):
%p A349782 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A349782 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 *)
%o A349782 (PARI) T(n, k) = sum(j=0, k, abs(stirling(n, j, 1))); \\ _Michel Marcus_, Dec 09 2021
%Y A349782 Row sums: A121586, central terms: A349783.
%Y A349782 Cf. A132393, A048994, A008275, A000142, A033312, A000774, A081046, A130534.
%K A349782 nonn,tabl
%O A349782 0,6
%A A349782 _Peter Luschny_, Dec 02 2021