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

%I A185778 #9 Jul 12 2017 03:18:53
%S A185778 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,
%T A185778 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,
%U A185778 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
%N A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.
%C A185778 Using "->" to mean "is the weight array of" as defined at A144112:
%C A185778 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.
%H A185778 G. C. Greubel, <a href="/A185778/b185778.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185778 (See the Mathematica code.)
%e A185778 Northwest corner:
%e A185778 1....-1....0....0....0....0....0,...0
%e A185778 -1....2....0....0....0....0....0....0
%e A185778 0.....0....0....1....1....1....1....1
%e A185778 0.....0....1....2....3....4....5....6
%e A185778 0.....0....1....3....6....10...15...21
%e A185778 0.....0....1....4....10...20...35...56
%t A185778 (* This code produces three arrays: A144225, A007318, A185778. *)
%t A185778 f[n_,0]:=0;f[0,k_]:=0;  (* Used to make the weight array *)
%t A185778 f[1,1]:=1;f[n_,1]:=0;f[1,k_]:=0
%t A185778 f[n_,2]:=1;f[2,k_]:=1;
%t A185778 f[n_,k_]:=-1+(n+k-4)!/((n-2)!*(k-2)!)/;k>1&&n>1;
%t A185778 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A144225 *)
%t A185778 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185778 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A007318, Pascal's triangle formatted as a rectangle *)
%t A185778 w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
%t A185778 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185778 *)
%t A185778 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185778 Cf. A144112, A185779, A144225, A007318, A014430, A077023, A185779.
%K A185778 sign,tabl
%O A185778 1,5
%A A185778 _Clark Kimberling_, Feb 03 2011