A130162 - OEIS

%I #15 Jun 27 2023 09:23:56

%S 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,

%T 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,

%U 1,1,2,3,0,7,0,0,0,0,0,63

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

%C Right border = A000837 (offset 1).

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

%F 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;...).

%F A051731 = inverse Moebius transform.

%e First few rows of the triangle:

%e   1;

%e   1,  1;

%e   1,  0,  2;

%e   1,  1,  0,  3;

%e   1,  0,  0,  0,  6;

%e   1,  1,  2,  0,  0,  7;

%e   1,  0,  0,  0,  0,  0, 14;

%e   1,  1,  0,  3,  0,  0,  0, 17;

%e   ...

%t 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 *)

%Y Cf. A000041, A000837, A051731.

%K nonn,tabl

%O 1,6

%A _Gary W. Adamson_, May 13 2007

%E More terms from _Jean-François Alcover_, Oct 03 2013

%E Offset changed to 1 by _Georg Fischer_, Jun 27 2023