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%I #28 Aug 05 2021 19:31:27
%S 0,0,129,2188,18700,94638,376761,1183920,3297456,8002300,18080425,
%T 37287660,73399404,135324378,241561425,410323648,680856256,1086411960,
%U 1703414961,2587286700,3877286700,5658888070,8172733129,11541726768,16164030000,22204797108
%N Sum of the seventh powers of the parts in the partitions of n into two distinct parts.
%H Colin Barker, <a href="/A294302/b294302.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_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1).
%F a(n) = Sum_{i=1..floor((n-1)/2)} i^7 + (n-i)^7.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8).
%F a(n) = (1/768)*(n^2*(64 - 224*n^2 + 448*n^4 - 3*(129 + (-1)^n)*n^5 + 96*n^6)).
%F a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - 28*a(n-4) + 28*a(n-5) + 56*a(n-6) - 56*a(n-7) - 70*a(n-8) + 70*a(n-9) + 56*a(n-10) - 56*a(n-11) - 28*a(n-12) + 28*a(n-13) + 8*a(n-14) - 8*a(n-15) - a(n-16) + a(n-17) for n>17.
%F (End)
%t Table[Sum[i^7 + (n - i)^7, {i, Floor[(n-1)/2]}], {n, 40}]
%t CoefficientList[Series[x^3(129+2059x+15480x^2+59466x^3+153639x^4+257307x^5+ 311664x^6+ 258532x^7+153639x^8+60537x^9+15480x^10+2178x^11+129x^12+x^13)/ ((1-x)^9 (1+x)^8),{x,0,60}],x] (* or *) LinearRecurrence[{1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1},{0,0,0,129,2188,18700,94638,376761,1183920,3297456,8002300,18080425,37287660,73399404,135324378,241561425,410323648},60] (* _Harvey P. Dale_, Aug 05 2021 *)
%o (PARI) a(n) = sum(i=1, (n-1)\2, i^7 + (n-i)^7); \\ _Michel Marcus_, Nov 08 2017
%o (PARI) concat(vector(2), Vec(x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%Y Cf. A294286, A294287, A294288, A294300, A294301.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 27 2017
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