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A345226 Irregular triangle, row sums equal A000041; in the format of A233932.

%I A345226 #34 Jun 20 2021 03:08:53
%S A345226 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,
%T A345226 7,30,17,23,7,54,17,23,7,54,51,23,7,95,51,23,7,95,51,23,7,55,161,51,
%U A345226 23,7,55,161,139,23,7,55,266,139,23,7,55,266,139,160,7,55,431,139,160,7,55
%N A345226 Irregular triangle, row sums equal A000041; in the format of A233932.
%C A345226 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.
%F A345226 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:
%F A345226   1
%F A345226   0, 1
%F A345226   1
%F A345226   0, 0, 1
%F A345226   1
%F A345226   0, 1
%F A345226   1
%F A345226   0, 0, 0, 1
%F A345226   ...
%F A345226 Replace the 1's with A187219 (the first difference row of A000041), getting M:
%F A345226   1
%F A345226   0, 1
%F A345226   1
%F A345226   0, 0, 2
%F A345226   2
%F A345226   0, 4
%F A345226   4
%F A345226   0, 0, 0, 7
%F A345226   ...
%F A345226 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.
%e A345226 The first few rows of the array equal A000041 with offset 1 as to sums:
%e A345226     1;
%e A345226     1,   1;
%e A345226     2,   1;
%e A345226     2,   1,  2;
%e A345226     4,   1,  2;
%e A345226     4,   5,  2;
%e A345226     8,   5,  2;
%e A345226     8,   5,  2, 7;
%e A345226    16,   5,  2, 7;
%e A345226    16,  17,  2, 7;
%e A345226    30,  17,  2, 7;
%e A345226    30,  17, 23, 7;
%e A345226    54,  17, 23, 7;
%e A345226    54,  51, 23, 7;
%e A345226    95,  51, 23, 7;
%e A345226    95,  51, 23, 7, 55;
%e A345226   161,  51, 23, 7, 55;
%e A345226   161, 139, 23, 7, 55;
%e A345226   ...
%e A345226 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, ...).
%Y A345226 Cf. A000041, A001511, A070939, A079559, A135764, A187219, A233932.
%K A345226 nonn,tabf
%O A345226 1,4
%A A345226 _Gary W. Adamson_, Jun 12 2021