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 A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 A086080 Adjacent sequences:  A185775 A185776 A185777 * A185779 A185780 A185781 KEYWORD sign,tabl AUTHOR Clark Kimberling, Feb 03 2011 STATUS approved

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)