%I #10 Feb 11 2018 10:55:33
%S 0,2,65,858,4890,21244,67171,188916,446964,994030,1978405,3796622,
%T 6735950,11680408,19092295,30745064,47260136,71929146,105409929,
%U 153455810,216455810,303993492,415601835,566623708,754740700,1003708134,1307797101,1702747126
%N Sum of the sixth powers of the parts in the partitions of n into two parts.
%H Colin Barker, <a href="/A294273/b294273.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).
%F a(n) = Sum_{i=1..floor(n/2)} i^6 + (n-i)^6.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7).
%F a(n) = (n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7).
%F a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
%F (End)
%t Table[Sum[i^6 + (n - i)^6, {i, Floor[n/2]}], {n, 50}]
%o (PARI) concat(0, Vec(x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Oct 26 2017
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