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

1,6

Comments

Right border = A000837 (offset 1).

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

Links

Table of n, a(n) for n=1..78.

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.

Example

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;
  ...

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 *)

Crossrefs

Cf. A000041, A000837, A051731.

Sequence in context: A152434 A143810 A128589 * A175595 A175417 A136481

Adjacent sequences: A130159 A130160 A130161 * A130163 A130164 A130165

Keyword

nonn,tabl

Author

Gary W. Adamson, May 13 2007

Extensions

More terms from Jean-François Alcover, Oct 03 2013

Offset changed to 1 by Georg Fischer, Jun 27 2023

Status

approved