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A185907
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Weight array of A185908, by descending antidiagonals.
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4
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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, 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, 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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1
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COMMENTS
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(See A144112 for definitions of weight array and accumulation array.)
Omitting the first row and reading the remaining array by descending antidiagonals results in A115356. - Georg Fischer, Jul 26 2023
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LINKS
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FORMULA
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All entries in column 1 and main diagonal are 1, all others are 0.
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EXAMPLE
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Northwest corner:
1, 0, 0, 0, 0, 0, 0, 0
1, 1, 0, 0, 0, 0, 0, 0
1, 0, 1, 0, 0, 0, 0, 0
1, 0, 0, 1, 0, 0, 0, 0
1, 0, 0, 0, 1, 0, 0, 0
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MATHEMATICA
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(* This program generates the array A185908, then its accumulation array, A185909, and then its weight array, A185907. *)
f[n_, 0]:=0; f[0, k_]:=0; (* needed for the weight array *)
f[n_, k_]:=Min[n, k]+n-1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
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}] (* A185909 *)]
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
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}]] (* A185907 *)
Table[w[n-k+1, k], {n, 16}, {k, n, 1, -1}]//Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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