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

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, 14, 16, 33, 71, 143, 9, 4, 9, 14, 16, 33, 71, 143, 295, 10, 5, 9, 14, 32, 33, 71, 143, 295, 594

Offset: 1

Author

Gary W. Adamson & Mats Granvik, Jul 30 2009

Keywords

• nonn
• tabl

Comments

This is an eigentriangle (i.e., a lower triangular matrix * a diagonalized version of its eigensequence); A014668 is the eigensequence of triangle A010766.

Row sums = A014668 starting (1, 3, 7, 16, 33, 71, 143, ...).

Sum of n-th row terms = rightmost term of next row.

Examples

			First few rows of the triangle =
   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, 14, 16, 33, 71, 143;
   9,  4,  9, 14, 16, 33, 71, 143, 295;
  10,  5,  9, 14, 32, 33, 71, 143, 295, 594;
  11,  5,  9, 14, 32, 33, 71, 143, 295, 594, 1206;
  12,  6, 12, 21, 32, 66, 71, 143, 295, 594, 1206, 2413;
  ...
Example: row 4 = (4, 2, 3, 7) = (4, 2, 1, 1) * (1, 1, 3, 7).
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A010766, A014668.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n -1}];
Table[Floor[n/k]* a[k], {n, 1, 5}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 18 2016 *)

Formula

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.