A309516 Number of odd parts in the partitions of n into 4 parts.

0, 0, 0, 0, 4, 3, 6, 7, 12, 14, 22, 25, 36, 40, 52, 59, 76, 85, 104, 116, 140, 154, 182, 200, 232, 254, 290, 316, 360, 389, 436, 471, 524, 564, 624, 669, 736, 786, 858, 915, 996, 1059, 1146, 1216, 1312, 1388, 1492, 1576, 1688, 1780, 1900, 2000, 2132, 2239

Offset: 0

Author

Wesley Ivan Hurt, Aug 05 2019

Keywords

• nonn

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     12          14          22          25          36        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions

Programs

Mathematica

Table[Sum[Sum[Sum[(Mod[i, 2] + Mod[j, 2] + Mod[k, 2] + Mod[n - i - j - k, 2]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]
Table[Count[Flatten[IntegerPartitions[n,{4}]],?OddQ],{n,0,60}] (* _Harvey P. Dale, Dec 30 2024 *)

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} ((i mod 2) + (j mod 2) + (k mod 2) + ((n-i-j-k) mod 2)).

Conjectures from Colin Barker, Aug 06 2019: (Start)

G.f.: x^4*(4 - 5*x + 4*x^2 - 2*x^3 + 4*x^4 - 3*x^5 + 2*x^6) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - a(n-16) for n>15.

(End)