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A163313 Triangle read by rows, A010766 convolved with A014668 (diagonalized as an infinite lower triangular matrix).

%I A163313 #16 Jun 02 2025 01:48:13
%S A163313 1,2,1,3,1,3,4,2,3,7,5,2,3,7,16,6,3,6,7,16,33,7,3,6,7,16,33,71,8,4,6,
%T A163313 14,16,33,71,143,9,4,9,14,16,33,71,143,295,10,5,9,14,32,33,71,143,295,
%U A163313 594
%N A163313 Triangle read by rows, A010766 convolved with A014668 (diagonalized as an infinite lower triangular matrix).
%C A163313 This is an eigentriangle (i.e., a lower triangular matrix * a diagonalized version of its eigensequence); A014668 is the eigensequence of triangle A010766.
%C A163313 Row sums = A014668 starting (1, 3, 7, 16, 33, 71, 143, ...).
%C A163313 Sum of n-th row terms = rightmost term of next row.
%H A163313 G. C. Greubel, <a href="/A163313/b163313.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A163313 Equals M * Q as infinite lower triangular matrices, where M = triangle A010766 and Q = a matrix with A014668: (1, 1, 3, 7, 16, 33, 71, 143, ...) as the main diagonal and the rest zeros.
%e A163313 First few rows of the triangle =
%e A163313    1;
%e A163313    2,  1;
%e A163313    3,  1,  3;
%e A163313    4,  2,  3,  7;
%e A163313    5,  2,  3,  7, 16;
%e A163313    6,  3,  6,  7, 16, 33;
%e A163313    7,  3,  6,  7, 16, 33  71;
%e A163313    8,  4,  6, 14, 16, 33, 71, 143;
%e A163313    9,  4,  9, 14, 16, 33, 71, 143, 295;
%e A163313   10,  5,  9, 14, 32, 33, 71, 143, 295, 594;
%e A163313   11,  5,  9, 14, 32, 33, 71, 143, 295, 594, 1206;
%e A163313   12,  6, 12, 21, 32, 66, 71, 143, 295, 594, 1206, 2413;
%e A163313   ...
%e A163313 Example: row 4 = (4, 2, 3, 7) = (4, 2, 1, 1) * (1, 1, 3, 7).
%t A163313 a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n -1}];
%t A163313 Table[Floor[n/k]* a[k], {n, 1, 5}, {k, 1, n}]//Flatten (* _G. C. Greubel_, Dec 18 2016 *)
%Y A163313 Cf. A010766, A014668.
%K A163313 nonn,tabl
%O A163313 1,2
%A A163313 _Gary W. Adamson_ & _Mats Granvik_, Jul 30 2009