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%I #21 Feb 02 2018 04:33:48
%S 0,0,65,730,4890,19786,67171,180724,446964,962780,1978405,3703310,
%T 6735950,11445110,19092295,30220776,47260136,70866264,105409929,
%U 151455810,216455810,300450370,415601835,560651740,754740700,994054516,1307797101,1687688054,2177107894
%N Sum of the sixth powers of the parts in the partitions of n into two distinct parts.
%H Colin Barker, <a href="/A294301/b294301.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-1)/2)} i^6 + (n-i)^6.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7).
%F a(n) = (n/42 - n^3/6 + n^5/2 - 1/128*(65 + (-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-1)/2]}], {n, 40}]
%o (PARI) a(n) = sum(i=1, (n-1)\2, i^6 + (n-i)^6); \\ _Michel Marcus_, Nov 08 2017
%o (PARI) concat(vector(2), Vec(x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%Y Cf. A294286, A294287, A294288, A294300.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 27 2017
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