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A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.

%I A185780 #14 Oct 01 2023 07:35:28
%S A185780 1,4,1,9,6,1,16,15,8,1,25,28,21,10,1,36,45,40,27,12,1,49,66,65,52,33,
%T A185780 14,1,64,91,96,85,64,39,16,1,81,120,133,126,105,76,45,18,1,100,153,
%U A185780 176,175,156,125,88,51,20,1,121,190,225,232,217,186,145,100,57,22,1,144,231,280,297,288,259,216,165,112,63,24,1,169,276,341,370,369,344,301,246,185,124,69,26,1,196,325,408,451,460,441,400,343,276,205,136,75,28,1
%N A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.
%C A185780 This is the accumulation array of A185781, the weight array of A185782, and second weight array of A185783.  See A144112 for definitions of accumulation array and weight array.
%H A185780 G. C. Greubel, <a href="/A185780/b185780.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185780 T(n,k) = k*(n*k - n + 1), k>=1, n>=1.
%e A185780 Northwest corner:
%e A185780   1....4....9....16....25....36
%e A185780   1....6....15...28....45....66
%e A185780   1....8....21...40....65....96
%e A185780   1....10...27...52....85....126
%t A185780 (* This code yields arrays A185780, A185781, and A185782. *)
%t A185780 f[n_,0]:=0;f[0,k_]:=0;  (* Used to make weight array A185782 *)
%t A185780 f[n_,k_]:=k(n*k-n+1);
%t A185780 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* this array *)
%t A185780 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185780 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc array of {f(n,k)} *)
%t A185780 FullSimplify[s[n,k]]
%t A185780 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A185781 *)
%t A185780 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185780 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 A185780 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* array A185782 *)
%t A185780 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* seq A185782 *)
%Y A185780 Cf. A144112, A185781, A185782, A185783.
%K A185780 nonn,tabl
%O A185780 1,2
%A A185780 _Clark Kimberling_, Feb 03 2011