%I #17 Sep 05 2019 02:31:48
%S 0,0,0,0,1,1,2,2,3,3,5,6,9,10,13,14,18,20,25,28,34,37,44,48,56,61,70,
%T 76,87,94,106,114,127,136,151,162,179,191,209,222,242,257,279,296,320,
%U 338,364,384,412,434,464,488,521,547,582,610,647,677,717,750
%N Number of odd parts appearing among the third largest parts of the partitions of n into 4 parts.
%H Colin Barker, <a href="/A309712/b309712.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,2,-2,1,0,0,0,-1,2,-1).
%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (j mod 2).
%F From _Colin Barker_, Aug 24 2019: (Start)
%F G.f.: x^4*(1 - x + x^2 - x^3 + x^4 - x^5 + 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) | 3 3 5 6 9 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 04 2019
%t Table[Sum[Sum[Sum[Mod[j, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
%t LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 6, 9, 10, 13, 14}, 60] (* _Wesley Ivan Hurt_, Sep 04 2019 *)
%o (PARI) concat([0,0,0,0], Vec(x^4*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)) + O(x^70))) \\ _Colin Barker_, Aug 24 2019
%Y Cf. A309711, A309715.
%K nonn,easy
%O 0,7
%A _Wesley Ivan Hurt_, Aug 13 2019