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%I #12 Sep 08 2019 01:58:24
%S 0,0,0,0,4,3,6,7,12,14,22,25,36,40,52,59,76,85,104,116,140,154,182,
%T 200,232,254,290,316,360,389,436,471,524,564,624,669,736,786,858,915,
%U 996,1059,1146,1216,1312,1388,1492,1576,1688,1780,1900,2000,2132,2239
%N Number of odd 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 mod 2) + (j mod 2) + (k mod 2) + ((n-i-j-k) mod 2)).
%F Conjectures from _Colin Barker_, Aug 06 2019: (Start)
%F 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)).
%F 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.
%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) | 12 14 22 25 36 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 07 2019
%t 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}]
%K nonn
%O 0,5
%A _Wesley Ivan Hurt_, Aug 05 2019
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