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%I #21 Sep 04 2019 17:27:09
%S 0,0,0,0,1,1,2,3,4,5,7,8,13,15,20,25,31,36,45,51,65,74,89,103,121,136,
%T 159,177,208,231,265,296,335,369,416,455,514,561,625,684,756,820,904,
%U 976,1076,1160,1268,1368,1488,1596,1732,1852,2009,2145,2314,2471
%N Sum of the odd parts appearing among the smallest parts of the partitions of n into 4 parts.
%H Colin Barker, <a href="/A309708/b309708.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_22">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1,0,2,-2,-2,0,2,2,-2,0,-1,1,1,0,-1,-1,1).
%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} k * (k mod 2).
%F From _Colin Barker_, Aug 23 2019: (Start)
%F G.f.: x^4*(1 + x^8) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2).
%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-12) + 2*a(n-13) - 2*a(n-14) - a(n-16) + a(n-17) + a(n-18) - a(n-20) - a(n-21) + a(n-22) for n > 21.
%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) | 4 5 7 8 13 ...
%e --------------------------------------------------------------------------
%t Table[Sum[Sum[Sum[k * Mod[k, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
%t LinearRecurrence[{1, 1, 0, -1, -1, 1, 0, 2, -2, -2, 0, 2, 2, -2, 0, -1, 1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 15, 20, 25, 31, 36, 45, 51, 65, 74}, 80] (* _Wesley Ivan Hurt_, Sep 04 2019 *)
%o (PARI) concat([0,0,0,0], Vec(x^4*(1 + x^8) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2) + O(x^60))) \\ _Colin Barker_, Aug 23 2019
%Y Cf. A309711, A309715.
%K nonn,easy
%O 0,7
%A _Wesley Ivan Hurt_, Aug 13 2019