A325773 Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 4), for k = 0, 1, 2, 3.

0, 1, 0, 0, 0, 2, 1, 0, 0, 4, 1, 1, 1, 7, 3, 1, 1, 13, 4, 2, 2, 20, 9, 4, 3, 32, 12, 7, 7, 48, 21, 10, 9, 73, 29, 17, 15, 106, 47, 24, 21, 153, 64, 37, 34, 215, 97, 53, 46, 303, 131, 76, 68, 416, 190, 106, 92, 571, 254, 151, 134, 770, 355, 204, 178, 1037

Offset: 1

Author

Clark Kimberling, Jun 05 2019

Keywords

• nonn
• tabf
• easy

Comments

Row n partitions A006128 into 4 parts, r(n,0) + r(n,1) + r(n,3) + r(n,4) = p(n) = A006128(n). What is the limiting behavior of r(n,0)/p(n)?

Examples

			First 15 rows:
   0      1      0      0
   0      2      1      0
   0      4      1      1
   1      7      3      1
   1     13      4      2
   2     20      9      4
   3     32     12      7
   7     48     21     10
   9     73     29     17
  15    106     47     24
  21    153     64     37
  34    215     97     53
  46    303    131     76
  68    416    190    106
  92    571    254    151
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A006128, A325771, A325772, A325774.

Programs

Mathematica

f[n_] := Mod[Flatten[IntegerPartitions[n]], 4];
Table[Count[f[n], k], {n, 1, 40}, {k,0,1,2,3}]  (* A325773 array *)
Flatten[%] (* A325773 sequence *)