A355260 - OEIS

%I #13 Jul 06 2022 11:13:58

%S 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 1275,780,203,0,720,3528,8120,11025,9100,4263,877,0,5040,26136,65660,

%U 101535,101920,65366,24556,4140,0,40320,219168,590620,1009260,1167348,920808,478842,149040,21147

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

%F T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^y - 1).

%F 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 T(n, k) = A000110(k) * A132393(n, k).

%e Triangle T(n, k) begins:

%e [0] 1;

%e [1] 0,   1;

%e [2] 0,   1,    2;

%e [3] 0,   2,    6,    5;

%e [4] 0,   6,   22,   30,   15;

%e [5] 0,  24,  100,  175,  150,    52;

%e [6] 0, 120,  548, 1125, 1275,   780,  203;

%e [7] 0, 720, 3528, 8120, 11025, 9100, 4263, 877;

%p Bell := n -> combinat[bell](n):

%p T := (n,k) -> Bell(k)*abs(Stirling1(n, k)):

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%p # Alternative:

%p egf := exp(1/(1 - x)^y - 1): ser := series(egf, x, 32):

%p cfx := n -> coeff(ser, x, n):

%p seq(seq(n!*coeff(cfx(n), y, k), k = 0..n), n = 0..8);

%t (* Utility function, extracts the lower triangular part of a square matrix. *)

%t TriangularForm[T_] := Table[Table[T[[n, k]], {k, 1, n}], {n, 1, Dimensions[T][[1]]}];

%t (* The actual calculation: *)

%t r := 9; R := Range[0, r];

%t T := Table[BellB[k] BellY[n, k, R!], {n, R}, {k, R}];

%t T // TriangularForm // Flatten

%Y Cf. A000262 (row sums), A033999 (alternating row sums), A000110 (main diagonal), A000142 (column 1).

%Y Cf. A132393, A355267.

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, Jul 06 2022