A130162 Triangle read by rows: A051731 * A000837 as a diagonalized matrix.

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 6, 1, 1, 2, 0, 0, 7, 1, 0, 0, 0, 0, 0, 14, 1, 1, 0, 3, 0, 0, 0, 17, 1, 0, 2, 0, 0, 0, 0, 0, 27, 1, 1, 0, 0, 6, 0, 0, 0, 0, 34, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 1, 1, 2, 3, 0, 7, 0, 0, 0, 0, 0, 63

Offset: 1

Author

Gary W. Adamson, May 13 2007

Keywords

• nonn
• tabl

Comments

Right border = A000837 (offset 1).

Row sums = partition numbers A000041 starting (1, 2, 3, 5, 7, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  0,  2;
  1,  1,  0,  3;
  1,  0,  0,  0,  6;
  1,  1,  2,  0,  0,  7;
  1,  0,  0,  0,  0,  0, 14;
  1,  1,  0,  3,  0,  0,  0, 17;
  ...
		

Crossrefs

Cf. A000041, A000837, A051731.

Programs

Mathematica

rows = 12; A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; A000837diag = DiagonalMatrix[Array[A000837, rows]]; A051731 = Table[ If[Mod[n, k] == 0, 1, 0], {n, 1, rows}, {k, 1, rows}]; A130162 = A051731.A000837diag; Table[ A130162[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

Formula

A051731 * A000837 (starting at offset 1) as a diagonalized matrix M, where M = T(n,k) = A000837(n) * 0^(n-k), 1<=k<=n; i.e., (1; 0,1; 0,0,2; 0,0,0,3; 0,0,0,0,6;...).

A051731 = inverse Moebius transform.

Extensions

More terms from Jean-François Alcover, Oct 03 2013

Offset changed to 1 by Georg Fischer, Jun 27 2023