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A144216
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C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.
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5
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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
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Table[n(n-m-1)+m(m+1)/2, {m, 15}, {n, m}] (* Paolo Xausa, Dec 21 2023 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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