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%I #34 Dec 21 2023 10:21:46
%S 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,
%T 28,22,18,16,16,18,22,28,36,29,24,21,20,21,24,29,36,45,37,31,27,25,25,
%U 27,31,37,45,55,46,39,34,31,30,31,34,39,46,55,66,56,48,42,38,36,36,38
%N C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.
%C This is the accumulation array (as defined at A144112) of the weight array A144217.
%C 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.
%H Paolo Xausa, <a href="/A144216/b144216.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened).
%F R(m,n) = (m(m-1)+n(n-1))/2.
%F 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
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
%e 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...
%e 3, 4, 6, 9, 13, 18, 24, 31, 39, 48, ...
%e 6, 7, 9, 12, 16, 21, 27, 34, 42, 51, ...
%e 10, 11, 13, 16, 20, 25, 31, 38, 46, 55, ...
%e 15, 16, 18, 21, 25, 30, 36, 43, 51, 60, ...
%e 21, 22, 24, 27, 31, 36, 42, 49, 57, 66, ...
%e 28, 29, 31, 34, 38, 43, 49, 56, 64, 73, ...
%e 36, 37, 39, 42, 46, 51, 57, 64, 72, 81, ...
%e 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, ...
%e R(2,4) = binomial(2,2) + binomial(4,2) = 1 + 6 = 7.
%p 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
%t Table[n(n-m-1)+m(m+1)/2,{m,15},{n,m}] (* _Paolo Xausa_, Dec 21 2023 *)
%Y Cf. A144217.
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_, Sep 14 2008
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