This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294275 #14 Jul 13 2025 11:09:10
%S A294275 0,2,257,7074,72354,469540,2142595,7972932,24684612,68121958,
%T A294275 167731333,383769830,812071910,1633567432,3103591687,5683259528,
%U A294275 9961449608,16980253770,27957167625,45040730666,70540730666,108577948908,163239463563,241980430540,351625763020
%N A294275 Sum of the eighth powers of the parts in the partitions of n into two parts.
%H A294275 Colin Barker, <a href="/A294275/b294275.txt">Table of n, a(n) for n = 1..1000</a>
%H A294275 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294275 <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 A294275 a(n) = Sum_{i=1..floor(n/2)} i^8 + (n-i)^8.
%F A294275 From _Colin Barker_, Nov 20 2017: (Start)
%F A294275 G.f.: x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9).
%F A294275 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 A294275 (End)
%F A294275 a(n) = -n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040. - _Wesley Ivan Hurt_, Jul 12 2025
%t A294275 Table[Sum[i^8 + (n - i)^8, {i, Floor[n/2]}], {n, 40}]
%o A294275 (PARI) concat(0, Vec( x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o A294275 (Magma) [-n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294275 Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), this sequence (k=8), A294276 (k=9), A294279 (k=10).
%K A294275 nonn,easy
%O A294275 1,2
%A A294275 _Wesley Ivan Hurt_, Oct 26 2017