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A355260 Triangle read by rows, T(n, k) = Bell(k) * |Stirling1(n, k)|.

%I A355260 #13 Jul 06 2022 11:13:58
%S A355260 1,0,1,0,1,2,0,2,6,5,0,6,22,30,15,0,24,100,175,150,52,0,120,548,1125,
%T A355260 1275,780,203,0,720,3528,8120,11025,9100,4263,877,0,5040,26136,65660,
%U A355260 101535,101920,65366,24556,4140,0,40320,219168,590620,1009260,1167348,920808,478842,149040,21147
%N A355260 Triangle read by rows, T(n, k) = Bell(k) * |Stirling1(n, k)|.
%F A355260 T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^y - 1).
%F A355260 T(n, k) = Bell(k)*Bell_{n, k}(A000142), where Bell_{n, k}(S) are the partial Bell polynomials mapped on the sequence S; here S are the factorial numbers. See the Mathematica program.
%F A355260 T(n, k) = A000110(k) * A132393(n, k).
%e A355260 Triangle T(n, k) begins:
%e A355260 [0] 1;
%e A355260 [1] 0,   1;
%e A355260 [2] 0,   1,    2;
%e A355260 [3] 0,   2,    6,    5;
%e A355260 [4] 0,   6,   22,   30,   15;
%e A355260 [5] 0,  24,  100,  175,  150,    52;
%e A355260 [6] 0, 120,  548, 1125, 1275,   780,  203;
%e A355260 [7] 0, 720, 3528, 8120, 11025, 9100, 4263, 877;
%p A355260 Bell := n -> combinat[bell](n):
%p A355260 T := (n,k) -> Bell(k)*abs(Stirling1(n, k)):
%p A355260 seq(seq(T(n, k), k = 0..n), n = 0..9);
%p A355260 # Alternative:
%p A355260 egf := exp(1/(1 - x)^y - 1): ser := series(egf, x, 32):
%p A355260 cfx := n -> coeff(ser, x, n):
%p A355260 seq(seq(n!*coeff(cfx(n), y, k), k = 0..n), n = 0..8);
%t A355260 (* Utility function, extracts the lower triangular part of a square matrix. *)
%t A355260 TriangularForm[T_] := Table[Table[T[[n, k]], {k, 1, n}], {n, 1, Dimensions[T][[1]]}];
%t A355260 (* The actual calculation: *)
%t A355260 r := 9; R := Range[0, r];
%t A355260 T := Table[BellB[k] BellY[n, k, R!], {n, R}, {k, R}];
%t A355260 T // TriangularForm // Flatten
%Y A355260 Cf. A000262 (row sums), A033999 (alternating row sums), A000110 (main diagonal), A000142 (column 1).
%Y A355260 Cf. A132393, A355267.
%K A355260 nonn,tabl
%O A355260 0,6
%A A355260 _Peter Luschny_, Jul 06 2022