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Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 3), for k = 0, 1, 2.
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%I #12 Aug 17 2020 13:42:14

%S 0,1,0,0,2,1,1,4,1,1,8,3,2,13,5,5,21,9,7,34,13,11,52,23,19,77,32,27,

%T 114,51,40,163,72,61,232,106,85,325,146,120,450,210,170,614,284,232,

%U 836,395,316,1120,529,433,1494,717,576,1976,946,767,2599,1264

%N Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 3), for k = 0, 1, 2.

%C Row n partitions A006128 into 3 parts, r(n,0) + r(n,1) + r(n,2) = p(n) = A006128(n). What is the limiting behavior of r(n,0)/p(n)?

%H Clark Kimberling, <a href="/A325772/b325772.txt">Table of n, a(n) for n = 1..150</a>

%e First 15 rows:

%e 0 1 0

%e 0 2 1

%e 1 4 1

%e 1 8 3

%e 2 13 5

%e 5 21 9

%e 7 34 13

%e 11 52 23

%e 19 77 32

%e 27 114 51

%e 40 163 72

%e 61 232 106

%e 85 325 146

%e 120 450 210

%e 170 614 264

%t f[n_] := Mod[Flatten[IntegerPartitions[n]], 3];

%t Table[Count[f[n], k], {n, 1, 40}, {k, 0, 1,2}] (* A325772 array *)

%t Flatten[%] (* A325772 sequence *)

%Y Cf. A006128, A325771, A325773, A325774.

%K nonn,tabf,easy

%O 1,5

%A _Clark Kimberling_, Jun 05 2019