A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

1, 0, 1, 0, 2, 0, 6, 6, 0, 24, 24, 0, 120, 240, 0, 720, 1440, 720, 0, 5040, 15120, 5040, 0, 40320, 120960, 80640, 0, 362880, 1451520, 1088640, 0, 3628800, 14515200, 14515200, 3628800, 0, 39916800, 199584000, 199584000, 39916800, 0, 479001600, 2395008000, 3353011200, 958003200

Offset: 0

Author

Geoffrey Critzer, Feb 23 2022

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2;
  0,     6,      6;
  0,    24,     24;
  0,   120,    240;
  0,   720,   1440,   720;
  0,  5040,  15120,  5040;
  0, 40320, 120960, 80640;
  ...
		

Crossrefs

Columns k=0-1 give: A000007, A000142 (for n>=1).

Cf. A088311 (row sums).

T(A000217(n),n) gives A052295.

Cf. A003056, A293140.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 23 2022

Mathematica

nn = 13; Prepend[Map[Prepend[#, 0] &, Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Product[1 + y x^i, {i, 1, nn}], {x, 0, nn}],{x,y}]], 1]], {1}] // Grid

Formula

E.g.f.: Product_{i>=1} (1 + y*x^i).

Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - Alois P. Heinz, Feb 23 2022