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A345226
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Irregular triangle, row sums equal A000041; in the format of A233932.
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1
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1, 1, 1, 2, 1, 2, 1, 2, 4, 1, 2, 4, 5, 2, 8, 5, 2, 8, 5, 2, 7, 16, 5, 2, 7, 16, 17, 2, 7, 30, 17, 2, 7, 30, 17, 23, 7, 54, 17, 23, 7, 54, 51, 23, 7, 95, 51, 23, 7, 95, 51, 23, 7, 55, 161, 51, 23, 7, 55, 161, 139, 23, 7, 55, 266, 139, 23, 7, 55, 266, 139, 160, 7, 55, 431, 139, 160, 7, 55
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OFFSET
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1,4
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COMMENTS
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The format of A233932 has a Gray code property of one term change in the next row. Using the production matrix shown below, we can obtain an array with row sums of any target sequence.
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LINKS
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FORMULA
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Let P equal the infinite lower triangular matrix with 1's in every row: [(1), (1, 1), (1, 1, 1), ...]. Begin with the following matrix format such that M[n, A001511(k)] = 1, otherwise 0:
1
0, 1
1
0, 0, 1
1
0, 1
1
0, 0, 0, 1
...
Replace the 1's with A187219 (the first difference row of A000041), getting M:
1
0, 1
1
0, 0, 2
2
0, 4
4
0, 0, 0, 7
...
Perform the operation P * M, getting A345226 as an irregular matrix. The operation P * M is equivalent to taking partial sums of column terms from top to bottom.
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EXAMPLE
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The first few rows of the array equal A000041 with offset 1 as to sums:
1;
1, 1;
2, 1;
2, 1, 2;
4, 1, 2;
4, 5, 2;
8, 5, 2;
8, 5, 2, 7;
16, 5, 2, 7;
16, 17, 2, 7;
30, 17, 2, 7;
30, 17, 23, 7;
54, 17, 23, 7;
54, 51, 23, 7;
95, 51, 23, 7;
95, 51, 23, 7, 55;
161, 51, 23, 7, 55;
161, 139, 23, 7, 55;
...
The leftmost column is (1, 1, 2, 2, 4, 4, 8, 8, ...), being the partial sums of the first column in matrix M: (1, 0, 1, 0, 2, 0, 4, ...).
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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