A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

1, -1, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0

Offset

1,5

Comments

Using "->" to mean "is the weight array of" as defined at A144112:

A185779->A144225->A007318->A014430->A077023->A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle). Read in reverse order, each is the accumulation array of the preceding array. It appears that successive weight arrays of A185779 contain Pascal's triangle except for initial terms.

Links

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

Formula

(See the Mathematica code.)

Example

Northwest corner:
1....-1....0....0....0....0....0,...0
-1....2....0....0....0....0....0....0
0.....0....0....1....1....1....1....1
0.....0....1....2....3....4....5....6
0.....0....1....3....6....10...15...21
0.....0....1....4....10...20...35...56

Mathematica

(* This code produces three arrays: A144225, A007318, A185778. *)
f[n_, 0]:=0; f[0, k_]:=0;  (* Used to make the weight array *)
f[1, 1]:=1; f[n_, 1]:=0; f[1, k_]:=0
f[n_, 2]:=1; f[2, k_]:=1;
f[n_, k_]:=-1+(n+k-4)!/((n-2)!*(k-2)!)/; k>1&&n>1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] (* A144225 *)
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}]] (* A007318, Pascal's triangle formatted as a rectangle *)
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}]] (* A185778 *)
Table[w[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

Crossrefs

Cf. A144112, A185779, A144225, A007318, A014430, A077023, A185779.

Sequence in context: A204843 A204853 A303709 * A071164 A027345 A328959

Adjacent sequences: A185775 A185776 A185777 * A185779 A185780 A185781

Keyword

sign,tabl

Author

Clark Kimberling, Feb 03 2011

Status

approved