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A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15,...). 4

%I

%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: (A000041 starts again in columns (1, 3, 7, 15,...)

%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

%F are A002865(a slight variant), A027336, A027338, A027342,...; i.e. the numbers

%F 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,tabl

%O 0,8

%A _Gary W. Adamson_, Feb 15 2010

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Last modified November 20 05:07 EST 2019. Contains 329323 sequences. (Running on oeis4.)