A309518 - OEIS

%I #13 Sep 08 2019 02:04:02

%S 0,0,0,0,0,1,2,5,8,10,14,19,24,32,40,49,60,71,84,100,116,134,154,176,

%T 200,226,254,284,316,351,388,429,472,516,564,615,668,726,786,849,916,

%U 985,1058,1136,1216,1300,1388,1480,1576,1676,1780,1888,2000,2117,2238

%N Number of even parts in the partitions of n into 4 parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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)).

%F Conjectures from _Colin Barker_, Aug 06 2019: (Start)

%F 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)).

%F 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).

%F (End)

%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..

%e                                                          1+1+1+9

%e                                                          1+1+2+8

%e                                                          1+1+3+7

%e                                                          1+1+4+6

%e                                              1+1+1+8     1+1+5+5

%e                                              1+1+2+7     1+2+2+7

%e                                  1+1+1+7     1+1+3+6     1+2+3+6

%e                                  1+1+2+6     1+1+4+5     1+2+4+5

%e                                  1+1+3+5     1+2+2+6     1+3+3+5

%e                      1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4

%e          1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6

%e          1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5

%e          1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4

%e          1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4

%e          2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3

%e --------------------------------------------------------------------------

%e   n  |      8           9          10          11          12        ...

%e --------------------------------------------------------------------------

%e a(n) |      8          10          14          19          24        ...

%e --------------------------------------------------------------------------

%e - _Wesley Ivan Hurt_, Sep 08 2019

%t 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}]

%t Table[Count[Flatten[IntegerPartitions[n,{4}]],_?EvenQ],{n,0,60}] (* _Harvey P. Dale_, Aug 20 2019 *)

%K nonn

%O 0,7

%A _Wesley Ivan Hurt_, Aug 05 2019