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A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).

%I A173304 #10 Feb 19 2022 08:56:47
%S A173304 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 A173304 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 A173304 1,66,96,93,40,2,88,121,120,53,3,105,154,154,72,5,137,193,196,94,7
%N A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).
%C A173304 Row sums = A000041, the partition numbers.
%F A173304 The generating array is in A173302.
%F A173304   1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...
%F A173304      1, 1, 2, 3, 5,  7, 11, 15, 22, 30, 42, 56,  77, 101, 135, ...
%F A173304            1, 1, 2,  3,  5,  7, 11, 15, 22, 30,  42,  56,  77, ...
%F A173304                          1,  1,  2,  3,  5,  7,  11,  15,  22, ...
%F A173304                                                             1, ...
%F A173304   ...
%F A173304 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 A173304 The finite difference array starts:
%e A173304   1, 1, 1, 1, 2, 2, 4, 4, 7,  8, 12, 14, 21, 24, ...; = A002865 (a variant)
%e A173304         1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
%e A173304            1, 1, 2, 3, 4, 6,  9, 12, 17, 23, 31, ...; = A017338
%e A173304                        1, 1,  2,  3,  5,  7, 11, ...; = A027342
%e A173304   ...
%e A173304 Last, columns of the array become rows of triangle A173304:
%e A173304     1;
%e A173304     1;
%e A173304     1,   1;
%e A173304     2,   2,   1;
%e A173304     2,   3,   2;
%e A173304     4,   4,   3;
%e A173304     4,   6,   4,  1;
%e A173304     7,   8,   6,  1;
%e A173304     8,  11,   9,  2;
%e A173304    12,  15,  12,  3;
%e A173304    14,  20,  17,  5;
%e A173304    21,  26,  23,  7;
%e A173304    24,  35,  31, 11;
%e A173304    34,  45,  41, 15;
%e A173304    41,  58,  55, 21, 1;
%e A173304    55,  75,  71, 29, 1;
%e A173304    66,  96,  93, 40, 2;
%e A173304    88, 121, 120, 53, 3;
%e A173304   105, 154, 154, 72, 5;
%e A173304   137, 193, 196, 94, 7;
%e A173304   ...
%Y A173304 Cf. A000041, A173301, A173302, A173303.
%K A173304 nonn,tabf,uned
%O A173304 0,8
%A A173304 _Gary W. Adamson_, Feb 15 2010