A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

0, 1, 1, 3, 2, 3, 6, 4, 4, 6, 10, 7, 6, 7, 10, 15, 11, 9, 9, 11, 15, 21, 16, 13, 12, 13, 16, 21, 28, 22, 18, 16, 16, 18, 22, 28, 36, 29, 24, 21, 20, 21, 24, 29, 36, 45, 37, 31, 27, 25, 25, 27, 31, 37, 45, 55, 46, 39, 34, 31, 30, 31, 34, 39, 46, 55, 66, 56, 48, 42, 38, 36, 36, 38

Offset: 1

Author

Clark Kimberling, Sep 14 2008

Keywords

• nonn
• tabl

Comments

This is the accumulation array (as defined at A144112) of the weight array A144217.

As a triangular array read by rows (0; 1, 1; 3, 2, 3; 6, 4, 4, 6; ...), T(n,j) = (1/2)n(n+1-2j)+j(j-1) (1<=j<=n) is the sum of the distances from the vertex j of the path graph 1-2-...-n to all the other vertices. Example: T(4,2)=4 because in the path 1-2-3-4 the distances from vertex 2 to the vertices 1, 2, 3, 4 are 1, 0, 1, 2, respectively; 1+0+1+2=4.

Examples

			   0,  1,  3,  6, 10, 15, 21, 28, 36, 45, ...
   1,  2,  4,  7, 11, 16, 22, 29, 37, 46, ...
   3,  4,  6,  9, 13, 18, 24, 31, 39, 48, ...
   6,  7,  9, 12, 16, 21, 27, 34, 42, 51, ...
  10, 11, 13, 16, 20, 25, 31, 38, 46, 55, ...
  15, 16, 18, 21, 25, 30, 36, 43, 51, 60, ...
  21, 22, 24, 27, 31, 36, 42, 49, 57, 66, ...
  28, 29, 31, 34, 38, 43, 49, 56, 64, 73, ...
  36, 37, 39, 42, 46, 51, 57, 64, 72, 81, ...
  45, 46, 48, 51, 55, 60, 66, 73, 81, 90, ...
R(2,4) = binomial(2,2) + binomial(4,2) = 1 + 6 = 7.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).

Crossrefs

Cf. A144217.

Programs

Maple

T := proc (n, j) if j <= n then (1/2)*n*(n+1-2*j)+j*(j-1) else 0 end if end proc: for n to 12 do seq(T(n, j), j = 1 .. n) end do; # yields sequence in triangular form

Mathematica

Table[n(n-m-1)+m(m+1)/2,{m,15},{n,m}] (* Paolo Xausa, Dec 21 2023 *)

Formula

R(m,n) = (m(m-1)+n(n-1))/2.

The sum of the terms in the upper left r X r submatrix is Sum_{n=1..r} Sum_{m=1..r} R(n,m) = A112742(r). - J. M. Bergot, Jun 18 2013