%I #16 Feb 05 2018 03:00:05
%S 0,0,513,19684,282340,2215782,12313161,52404624,186884496,572351860,
%T 1574304985,3922174980,9092033028,19656178794,40357579185,78666720832,
%U 147520415296,265720871304,464467582161,786155279940,1299155279940,2091077378830,3300704544313
%N Sum of the ninth powers of the parts of the partitions of n into two distinct parts.
%H Colin Barker, <a href="/A294304/b294304.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_21">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).
%F a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10).
%F a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
%F (End)
%t Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}]
%o (PARI) a(n) = sum(i=1, (n-1)\2, i^9 + (n-i)^9); \\ _Michel Marcus_, Nov 08 2017
%o (PARI) concat(vector(2), Vec(x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%Y Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 27 2017
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