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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Feb 06 2011

Keywords

• nonn
• tabl

Comments

A member of the accumulation chain

...< A185916 < A185914 < A185915 < ...

(See A144112 for definitions of weight array and accumulation array.)

Examples

			Northwest corner:
1...2...3...4...5...6...7...8...9
0...1...2...3...4...5...6...7...8
0...0...1...2...3...4...5...6...7
0...0...0...1...2...3...4...5...6
0...0...0...0...1...2...3...4...5
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A144112, A185915, A185916.

Programs

Mathematica

(* This program generates the array A185914, its accumulation array A185915, and its weight array A185916. *)
f[n_,0]:=0;f[0,k_]:=0; (* needed for the weight array *)
f[n_,k_]:=k-n+1; f[n_,k_]:=0/;kA185914 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A184915 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A184916 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

Formula

T(n,k) = k-n+1 for k>=n; T(n,k)=0 for k=1, n>=1.