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Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.
2

%I #9 Jul 12 2017 03:18:53

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

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

%C 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 G. C. Greubel, <a href="/A185778/b185778.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F (See the Mathematica code.)

%e Northwest corner:

%e 1....-1....0....0....0....0....0,...0

%e -1....2....0....0....0....0....0....0

%e 0.....0....0....1....1....1....1....1

%e 0.....0....1....2....3....4....5....6

%e 0.....0....1....3....6....10...15...21

%e 0.....0....1....4....10...20...35...56

%t (* This code produces three arrays: A144225, A007318, A185778. *)

%t f[n_,0]:=0;f[0,k_]:=0; (* Used to make the weight array *)

%t f[1,1]:=1;f[n_,1]:=0;f[1,k_]:=0

%t f[n_,2]:=1;f[2,k_]:=1;

%t f[n_,k_]:=-1+(n+k-4)!/((n-2)!*(k-2)!)/;k>1&&n>1;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A144225 *)

%t s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A007318, Pascal's triangle formatted as a rectangle *)

%t 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 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185778 *)

%t Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

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

%K sign,tabl

%O 1,5

%A _Clark Kimberling_, Feb 03 2011