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A144216
C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.
5
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
1,4
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.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).
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
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
MATHEMATICA
Table[n(n-m-1)+m(m+1)/2, {m, 15}, {n, m}] (* Paolo Xausa, Dec 21 2023 *)
CROSSREFS
Cf. A144217.
Sequence in context: A266593 A130459 A323621 * A127938 A131990 A033771
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Sep 14 2008
STATUS
approved