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A185914 Array: T(n,k)=k-n+1 for k>=n; T(n,k)=0 for k by antidiagonals.

%I A185914 #7 Jul 22 2017 17:28:38
%S A185914 1,2,0,3,1,0,4,2,0,0,5,3,1,0,0,6,4,2,0,0,0,7,5,3,1,0,0,0,8,6,4,2,0,0,
%T A185914 0,0,9,7,5,3,1,0,0,0,0,10,8,6,4,2,0,0,0,0,0,11,9,7,5,3,1,0,0,0,0,0,12,
%U A185914 10,8,6,4,2,0,0,0,0,0,0,13,11,9,7,5,3,1,0,0,0,0,0,0,14,12,10,8,6,4,2,0,0,0,0,0,0,0
%N A185914 Array: T(n,k)=k-n+1 for k>=n; T(n,k)=0 for k<n; by antidiagonals.
%C A185914 A member of the accumulation chain
%C A185914 ...< A185916 < A185914 < A185915 < ...
%C A185914 (See A144112 for definitions of weight array and accumulation array.)
%H A185914 G. C. Greubel, <a href="/A185914/b185914.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185914 T(n,k) = k-n+1 for k>=n; T(n,k)=0 for k<n; k>=1, n>=1.
%e A185914 Northwest corner:
%e A185914 1...2...3...4...5...6...7...8...9
%e A185914 0...1...2...3...4...5...6...7...8
%e A185914 0...0...1...2...3...4...5...6...7
%e A185914 0...0...0...1...2...3...4...5...6
%e A185914 0...0...0...0...1...2...3...4...5
%t A185914 (* This program generates the array A185914, its accumulation array A185915, and its weight array A185916. *)
%t A185914 f[n_,0]:=0;f[0,k_]:=0; (* needed for the weight array *)
%t A185914 f[n_,k_]:=k-n+1; f[n_,k_]:=0/;k<n;
%t A185914 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* A185914 *)
%t A185914 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185914 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
%t A185914 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A184915 *)
%t A185914 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185914 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 A185914 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A184916 *)
%t A185914 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185914 Cf. A144112, A185915, A185916.
%K A185914 nonn,tabl
%O A185914 1,2
%A A185914 _Clark Kimberling_, Feb 06 2011