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

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

1,2

Comments

A member of the accumulation chain

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

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

Links

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

Formula

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

Example

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

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/; k<n;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]  (* A185914 *)
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

Crossrefs

Cf. A144112, A185915, A185916.

Sequence in context: A127374 A098862 A003988 * A144257 A257232 A321980

Adjacent sequences: A185911 A185912 A185913 * A185915 A185916 A185917

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 06 2011

Status

approved