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

1,2

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.

Links

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

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.

Example

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).

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 *)

Crossrefs

Cf. A010766, A014668.

Sequence in context: A143802 A177040 A336887 * A337178 A321893 A308673

Adjacent sequences: A163310 A163311 A163312 * A163314 A163315 A163316

Keyword

nonn,tabl

Author

Gary W. Adamson & Mats Granvik, Jul 30 2009

Status

approved