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 A294271 #19 Jul 13 2025 11:06:48
%S A294271 0,2,17,114,354,1060,2275,4932,8772,15958,25333,41270,60710,91672,
%T A294271 127687,182408,243848,333930,432345,572666,722666,931788,1151403,
%U A294271 1451980,1763020,2182206,2610621,3180478,3756718,4514624,5273999,6263056,7246096,8515538,9768353
%N A294271 Sum of the fourth powers of the parts in the partitions of n into two parts.
%H A294271 Colin Barker, <a href="/A294271/b294271.txt">Table of n, a(n) for n = 1..1000</a>
%H A294271 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294271 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).
%F A294271 a(n) = Sum_{i=1..floor(n/2)} i^4 + (n-i)^4.
%F A294271 From _Colin Barker_, Nov 20 2017: (Start)
%F A294271 G.f.: x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5).
%F A294271 a(n) = (1/480)*(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4)).
%F A294271 a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>11.
%F A294271 (End)
%t A294271 Table[Sum[i^4 + (n - i)^4, {i, Floor[n/2]}], {n, 60}]
%t A294271 Table[Total[Flatten[IntegerPartitions[n,{2}]]^4],{n,40}] (* _Harvey P. Dale_, Mar 01 2019 *)
%o A294271 (PARI) concat(0, Vec(x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o A294271 (PARI) a(n) = sum(i=1, n\2, i^4 + (n-i)^4); \\ _Michel Marcus_, Nov 20 2017
%o A294271 (Magma) [(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4))/480 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294271 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), this sequence (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
%K A294271 nonn,easy
%O A294271 1,2
%A A294271 _Wesley Ivan Hurt_, Oct 26 2017