A309518 Number of even parts in the partitions of n into 4 parts.

0, 0, 0, 0, 0, 1, 2, 5, 8, 10, 14, 19, 24, 32, 40, 49, 60, 71, 84, 100, 116, 134, 154, 176, 200, 226, 254, 284, 316, 351, 388, 429, 472, 516, 564, 615, 668, 726, 786, 849, 916, 985, 1058, 1136, 1216, 1300, 1388, 1480, 1576, 1676, 1780, 1888, 2000, 2117, 2238

Offset

0,7

Links

Table of n, a(n) for n=0..54.

Index entries for sequences related to partitions

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-1) mod 2) + ((j-1) mod 2) + ((k-1) mod 2) + ((n-i-j-k-1) mod 2)).

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

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

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 3*a(n-6) + a(n-7) + a(n-8) - 3*a(n-9) + 4*a(n-10) - 4*a(n-11) + 4*a(n-12) - 4*a(n-13) + 3*a(n-14) - a(n-15).

(End)

Example

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) |      8          10          14          19          24        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 08 2019

Mathematica

Table[Sum[Sum[Sum[(Mod[i - 1, 2] + Mod[j - 1, 2] + Mod[k - 1, 2] + Mod[n - i - j - k - 1, 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}]], _?EvenQ], {n, 0, 60}] (* Harvey P. Dale, Aug 20 2019 *)

Crossrefs

Sequence in context: A189457 A189362 A189928 * A169922 A157481 A100809

Adjacent sequences: A309515 A309516 A309517 * A309519 A309520 A309521

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 05 2019

Status

approved