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 A294274 #14 Jul 13 2025 11:08:44
%S A294274 0,2,129,2444,18700,99012,376761,1216688,3297456,8158550,18080425,
%T A294274 37847532,73399404,136971464,241561425,414517952,680856256,1095977898,
%U A294274 1703414961,2607286700,3877286700,5697862412,8172733129,11613390384,16164030000,22330294142
%N A294274 Sum of the seventh powers of the parts in the partitions of n into two parts.
%H A294274 Colin Barker, <a href="/A294274/b294274.txt">Table of n, a(n) for n = 1..1000</a>
%H A294274 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294274 <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 A294274 a(n) = Sum_{i=1..floor(n/2)} i^7 + (n-i)^7.
%F A294274 From _Colin Barker_, Nov 20 2017: (Start)
%F A294274 G.f.: x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8).
%F A294274 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 A294274 (End)
%F A294274 a(n) = n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768. - _Wesley Ivan Hurt_, Jul 12 2025
%t A294274 Table[Sum[i^7 + (n - i)^7, {i, Floor[n/2]}], {n, 40}]
%o A294274 (PARI) concat(0, Vec(x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o A294274 (Magma) [n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294274 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), this sequence (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
%K A294274 nonn,easy
%O A294274 1,2
%A A294274 _Wesley Ivan Hurt_, Oct 26 2017