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A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

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