A355260 Triangle read by rows, T(n, k) = Bell(k) * |Stirling1(n, k)|.
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, 1275, 780, 203, 0, 720, 3528, 8120, 11025, 9100, 4263, 877, 0, 5040, 26136, 65660, 101535, 101920, 65366, 24556, 4140, 0, 40320, 219168, 590620, 1009260, 1167348, 920808, 478842, 149040, 21147
Offset: 0
Examples
Triangle T(n, k) begins: [0] 1; [1] 0, 1; [2] 0, 1, 2; [3] 0, 2, 6, 5; [4] 0, 6, 22, 30, 15; [5] 0, 24, 100, 175, 150, 52; [6] 0, 120, 548, 1125, 1275, 780, 203; [7] 0, 720, 3528, 8120, 11025, 9100, 4263, 877;
Crossrefs
Programs
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Maple
Bell := n -> combinat[bell](n): T := (n,k) -> Bell(k)*abs(Stirling1(n, k)): seq(seq(T(n, k), k = 0..n), n = 0..9); # Alternative: egf := exp(1/(1 - x)^y - 1): ser := series(egf, x, 32): cfx := n -> coeff(ser, x, n): seq(seq(n!*coeff(cfx(n), y, k), k = 0..n), n = 0..8);
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Mathematica
(* Utility function, extracts the lower triangular part of a square matrix. *) TriangularForm[T_] := Table[Table[T[[n, k]], {k, 1, n}], {n, 1, Dimensions[T][[1]]}]; (* The actual calculation: *) r := 9; R := Range[0, r]; T := Table[BellB[k] BellY[n, k, R!], {n, R}, {k, R}]; T // TriangularForm // Flatten