A283368 Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.

1, 2, 3, 2, 4, 3, 5, 3, 6, 5, 4, 7, 5, 4, 8, 6, 5, 9, 7, 5, 10, 8, 7, 6, 11, 8, 7, 6, 12, 10, 9, 7, 13, 10, 9, 7, 14, 11, 9, 8, 15, 12, 11, 10, 8, 16, 13, 12, 11, 9, 17, 13, 12, 11, 9, 18, 15, 13, 12, 10, 19, 15, 13, 12, 10, 20, 16, 15, 13, 11, 21, 17, 16, 15, 14, 11

Offset: 1

Author

Hartmut F. W. Hoft, Mar 06 2017

Keywords

• nonn
• tabf

Comments

The dot product of the n-th row of this triangle and the n-th row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the x-axis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.

Examples

			The first horizontal leg of the symmetric representation of sigma(15) is at y-coordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.
The first 16 rows of the irregular triangle:
   1
   2
   3    2
   4    3
   5    3
   6    5    4
   7    5    4
   8    6    5
   9    7    5
  10    8    7    6
  11    8    7    6
  12   10    9    7
  13   10    9    7
  14   11    9    8
  15   12   11   10    8
  16   13   12   11    9
		

Crossrefs

Cf. A024916, A237593, A259176, A259177, A283367.

Programs

Mathematica

(* function f[n,k] and its support functions are defined in A237593 *)
a283368[n_, k_] := n - Sum[f[n, 2i], {i, k-1}]
TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)

Formula

T(n,k) = n - Sum_{i=1..k-1} f(n, 2*i) where f is defined in A237593.

A024916(n) = Sum_{i=1..row(n)} T(n,i)*S(n,i) where S(n,i) refers to the triangle of A259176 and row(n) = floor((sqrt(8*n+1)-1)/2).