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%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
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