A185907 Weight array of A185908, by descending antidiagonals.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1

Comments

A member of the accumulation chain ... < A185907 < A185908 < A185909 < ...

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

Omitting the first row and reading the remaining array by descending antidiagonals results in A115356. - Georg Fischer, Jul 26 2023

Links

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

Formula

All entries in column 1 and main diagonal are 1, all others are 0.

Example

Northwest corner:
  1, 0, 0, 0, 0, 0, 0, 0
  1, 1, 0, 0, 0, 0, 0, 0
  1, 0, 1, 0, 0, 0, 0, 0
  1, 0, 0, 1, 0, 0, 0, 0
  1, 0, 0, 0, 1, 0, 0, 0

Mathematica

(* This program generates the array A185908, then its accumulation array, A185909, and then its weight array, A185907. *)
f[n_, 0]:=0; f[0, k_]:=0;  (* needed for the weight array *)
f[n_, k_]:=Min[n, k]+n-1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
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}]; (* accumulation array of {f(n, k)} *)
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]  (* A185909 *)]
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}]] (* A185907 *)
Table[w[n-k+1, k], {n, 16}, {k, n, 1, -1}]//Flatten

Crossrefs

Cf. A115356, A144112, A185908, A185909.

Sequence in context: A355655 A077052 A133566 * A321016 A077051 A115955

Adjacent sequences: A185904 A185905 A185906 * A185908 A185909 A185910

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 06 2011

Status

approved