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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 (list; table; graph; refs; listen; history; text; internal format)
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+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.

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(m-1)+n(n-1))/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+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

CROSSREFS

Cf. A144217.

Sequence in context: A220075 A266593 A130459 * A127938 A131990 A033771

Adjacent sequences:  A144213 A144214 A144215 * A144217 A144218 A144219

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Sep 14 2008

STATUS

approved

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Last modified February 18 13:07 EST 2018. Contains 299322 sequences. (Running on oeis4.)