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A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

%I A185877 #14 Jul 20 2017 02:19:01
%S A185877 1,3,1,7,5,1,13,11,7,1,21,19,15,9,1,31,29,25,19,11,1,43,41,37,31,23,
%T A185877 13,1,57,55,51,45,37,27,15,1,73,71,67,61,53,43,31,17,1,91,89,85,79,71,
%U A185877 61,49,35,19,1,111,109,105,99,91,81,69,55,39,21,1,133,131,127,121,113,103,91,77,61,43,23,1,157,155,151,145,137,127,115,101,85,67,47,25,1,183,181,177,171,163,153,141,127,111,93,73,51,27,1
%N A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.
%C A185877 A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).
%H A185877 G. C. Greubel, <a href="/A185877/b185877.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185877 T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.
%e A185877 Northwest corner:
%e A185877   1, 3,  7, 13, 21
%e A185877   1, 5, 11, 19, 29
%e A185877   1, 7, 15, 25, 45
%e A185877   1, 9, 19, 31, 45
%t A185877 (* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)
%t A185877 f[n_,0]:=0;f[0,k_]:=0;
%t A185877 f[n_,k_]:=k^2+(2n-3)k-2n+3;
%t A185877 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185877 *)
%t A185877 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185877 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185877 FullSimplify[s[n,k]]  (* formula for A185878 *)
%t A185877 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
%t A185877 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185877 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 A185877 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185879 *)
%t A185877 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185877 Cf. A144112, A185878, A185879, A185880.
%Y A185877 Row 1 to 3: A002061, A028387, A082111.
%Y A185877 diag (1,5,...): A056108;
%Y A185877 diag (3,11,...): A056106;
%Y A185877 diag (7,19,...): A003215;
%Y A185877 diag (13,29,...): A144391;
%Y A185877 diag (1,7,...): A003215;
%Y A185877 diag (1,9,...): A144390.
%K A185877 nonn,tabl
%O A185877 1,2
%A A185877 _Clark Kimberling_, Feb 05 2011