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%I #24 Jul 31 2023 16:26:03
%S 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,
%T 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,
%U 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
%N Weight array of A185908, by descending antidiagonals.
%C A member of the accumulation chain ... < A185907 < A185908 < A185909 < ...
%C (See A144112 for definitions of weight array and accumulation array.)
%C Omitting the first row and reading the remaining array by descending antidiagonals results in A115356. - _Georg Fischer_, Jul 26 2023
%H G. C. Greubel, <a href="/A185907/b185907.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F All entries in column 1 and main diagonal are 1, all others are 0.
%e Northwest corner:
%e 1, 0, 0, 0, 0, 0, 0, 0
%e 1, 1, 0, 0, 0, 0, 0, 0
%e 1, 0, 1, 0, 0, 0, 0, 0
%e 1, 0, 0, 1, 0, 0, 0, 0
%e 1, 0, 0, 0, 1, 0, 0, 0
%t (* This program generates the array A185908, then its accumulation array, A185909, and then its weight array, A185907. *)
%t f[n_,0]:=0;f[0,k_]:=0; (* needed for the weight array *)
%t f[n_,k_]:=Min[n,k]+n-1;
%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%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}] (* A185909 *)]
%t Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%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}]] (* A185907 *)
%t Table[w[n-k+1,k],{n,16},{k,n,1,-1}]//Flatten
%Y Cf. A115356, A144112, A185908, A185909.
%K nonn,tabl
%O 1
%A _Clark Kimberling_, Feb 06 2011
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