

A144216


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


2



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
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OFFSET

1,4


COMMENTS

R 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+12j)+j(j1) (1<=j<=n) is the sum of the distances from the vertex j of the path graph 12...n to all the other vertices. Example: T(4,2)=4 because in the path 1234 the distances from vertex 2 to the vertices 1, 2, 3, 4 are 1, 0, 1, 2, respectively; 1+0+1+2=4.
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


LINKS

Table of n, a(n) for n=1..74.


FORMULA

R(m,n) = (m(m1)+n(n1))/2.


EXAMPLE

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.


MAPLE

T := proc (n, j) if j <= n then (1/2)*n*(n+12*j)+j*(j1) 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


CROSSREFS

Cf. A144217.
Sequence in context: A266593 A130459 A323621 * A127938 A131990 A033771
Adjacent sequences: A144213 A144214 A144215 * A144217 A144218 A144219


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Sep 14 2008


STATUS

approved



