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%I #25 Feb 05 2018 02:13:51
%S 0,0,257,6562,72354,456418,2142595,7841860,24684612,67340708,
%T 167731333,380410598,812071910,1622037830,3103591687,5649705096,
%U 9961449608,16894160328,27957167625,44840730666,70540730666,108149231146,163239463563,241120467148,351625763020
%N Sum of the eighth powers of the parts in the partitions of n into two distinct parts.
%H David A. Corneth, <a href="/A294303/b294303.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1,9,-9,-36,36,84,-84,-126,126,126,-126,-84,84,36,-36,-9,9,1,-1).
%F a(n) = Sum_{i=1..floor((n-1)/2)} i^8 + (n-i)^8.
%F From _David A. Corneth_, Nov 05 2017: (Start)
%F For odd n, a(n) = n^9 / 9 - n^8/2 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30
%F For even n, a(n) = n^9 / 9 - 129*n^8/256 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30.
%F (End)
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9).
%F a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.
%F (End)
%t Table[Sum[i^8 + (n - i)^8, {i, Floor[(n-1)/2]}], {n, 30}]
%o (PARI) first(n) = {my(res = vector(n, i, 1/9*i^9 - 1/2*i^8 + 2/3*i^7 - 7/15*i^5 + 2/9*i^3 - 1/30*i)); forstep(i = 2, #res, 2, res[i] -= i^8/256); res} \\ _David A. Corneth_, Nov 05 2017
%o (PARI) concat(vector(2), Vec(x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%Y Cf. A294286, A294287, A294288, A294300, A294301, A294302.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 27 2017
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