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%I #10 Feb 19 2022 08:56:47
%S 1,1,1,1,1,1,1,2,2,1,2,3,2,4,4,3,4,6,4,7,8,6,1,8,11,9,2,12,15,12,3,14,
%T 20,17,5,21,26,23,7,24,35,31,11,34,45,41,15,41,58,55,21,1,55,75,71,29,
%U 1,66,96,93,40,2,88,121,120,53,3,105,154,154,72,5,137,193,196,94,7
%N Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).
%C Row sums = A000041, the partition numbers.
%F The generating array is in A173302.
%F 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...
%F 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, ...
%F 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
%F 1, 1, 2, 3, 5, 7, 11, 15, 22, ...
%F 1, ...
%F ...
%F Take finite differences from the bottom, creating a new array in which rows are A002865 (a slight variant), A027336, A027338, A027342, ...; i.e., the numbers of partitions of n that do not contain (1, 2, 4, 8, ...) as a part.
%e The finite difference array starts:
%e 1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, ...; = A002865 (a variant)
%e 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
%e 1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, ...; = A017338
%e 1, 1, 2, 3, 5, 7, 11, ...; = A027342
%e ...
%e Last, columns of the array become rows of triangle A173304:
%e 1;
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 2, 3, 2;
%e 4, 4, 3;
%e 4, 6, 4, 1;
%e 7, 8, 6, 1;
%e 8, 11, 9, 2;
%e 12, 15, 12, 3;
%e 14, 20, 17, 5;
%e 21, 26, 23, 7;
%e 24, 35, 31, 11;
%e 34, 45, 41, 15;
%e 41, 58, 55, 21, 1;
%e 55, 75, 71, 29, 1;
%e 66, 96, 93, 40, 2;
%e 88, 121, 120, 53, 3;
%e 105, 154, 154, 72, 5;
%e 137, 193, 196, 94, 7;
%e ...
%Y Cf. A000041, A173301, A173302, A173303.
%K nonn,tabf,uned
%O 0,8
%A _Gary W. Adamson_, Feb 15 2010
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