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A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.

%I A185910 #24 Oct 25 2021 08:20:48
%S A185910 1,2,4,3,5,9,4,6,10,16,5,7,11,17,25,6,8,12,18,26,36,7,9,13,19,27,37,
%T A185910 49,8,10,14,20,28,38,50,64,9,11,15,21,29,39,51,65,81,10,12,16,22,30,
%U A185910 40,52,66,82,100,11,13,17,23,31,41,53,67,83,101,121,12,14,18,24,32,42,54,68,84,102,122,144,13,15,19,25,33,43,55,69,85,103,123,145,169,14,16,20,26,34,44,56,70,86,104,124,146,170,196
%N A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.
%C A185910 A member of the accumulation chain ... < A185911 < A185910 < A185912 < A185913 < ... (See A144112 for definitions of weight array and accumulation array.)
%H A185910 G. C. Greubel, <a href="/A185910/b185910.txt">Table of n, a(n) for the first 50 antidiagonals, flattened</a>
%F A185910 T(n,k) = n^2 + k - 1, k >= 1, n >= 1.
%e A185910 Northwest corner:
%e A185910    1,  2,  3,  4,  5
%e A185910    4,  5,  6,  7,  8
%e A185910    9, 10, 11, 12, 13
%e A185910   16, 17, 18, 19, 20
%t A185910 (* This program generates the array A185910, its accumulation array A185812, and its weight array A185911. *)
%t A185910 f[n_,0]:=0;f[0,k_]:=0;
%t A185910 f[n_,k_]:=n^2+k-1;
%t A185910 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185910 *)
%t A185910 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185910 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185910 FullSimplify[s[n,k]]  (* formula for A185812 *)
%t A185910 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
%t A185910 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185910 w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
%t A185910 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185911 *)
%t A185910 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185910 Cf. A144112, A185910, A185912.
%K A185910 nonn,tabl
%O A185910 1,2
%A A185910 _Clark Kimberling_, Feb 06 2011