A144112 Weight array W={w(i,j)} of the natural number array A000027.

1, 1, 2, 2, 1, 3, 3, 1, 1, 4, 4, 1, 1, 1, 5, 5, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 1, 1, 1, 1, 1, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Sep 11 2008

Keywords

• nonn
• tabl

Comments

The lattice lines in the first quadrant (including the x and y axes) cut the plane into unit squares. Suppose a weight w(i,j) is assigned to the square that has as upper right corner the point (i,j). Let s(m,n) be the sum of the weights w(i,j) for 1<=i<=m, 1<=j<=n. We call the array W={w(i,j)} the weight array of the array S={s(m,n)} and S the accumulation array of W. For the case at hand, S is the array of natural numbers having the following antidiagonals: (1), then (2,3), then (4,5,6), then (7,8,9,10) and so on.

Contribution from Clark Kimberling, Sep 14 2008: (Start)

In general, the weight array W of an arbitrary rectangular array S={s(i,j): i>=1, j>=1} is defined in two steps:

(1) extend s by defining s(i,j)=0 if i=0 or j=0;

(2) then w(m,n)=s(m,n)+s(m-1,n-1)-s(m,n-1)-s(m-1,n) for m>=1, n>=1. (End)

Examples

			From _Clark Kimberling_, Jan 31 2011: (Start)
Northwest corner:
  1 1 2 3 4 5
  2 1 1 1 1 1
  3 1 1 1 1 1
  4 1 1 1 1 1
  5 1 1 1 1 1.
so that the accumulation array has corner:
  1...2...4...7...11...16
  3...5...8...12..17...23
  6...9...13..18..24...31
  10..14..19..25..32...40
  15..20..26..33..41...50.
s(2,4)=1+1+2+3+2+1+1+1=12. (End)
		

Links

• Stefano Spezia, Table of n, a(n) for n = 1..11325, (first 150 antidiagonals, flattened).

Crossrefs

Cf. A000012, A000027.

Programs

Mathematica

Array[Append[PadRight[{#},#,1],#+1]&,15,0] (* Paolo Xausa, Dec 21 2023 *)

Formula

Row 1: 1 followed by A000027.

Row n: n followed by A000012, for n>1.

G.f.: x*y*(1 - (1 + x)*y + (1 - x + x^2)*y^2)/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Oct 01 2023