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 A294279 #16 Jul 13 2025 11:10:08
%S A294279 0,2,1025,61098,1108650,10933324,71340451,354864276,1427557524,
%T A294279 4924107550,14914341925,40912232702,102769130750,240910097848,
%U A294279 529882277575,1107606410024,2206044295976,4225524980826,7792505423049,13933571680850,24163571680850,40869390083652
%N A294279 Sum of the tenth powers of the parts in the partitions of n into two parts.
%H A294279 Robert Israel, <a href="/A294279/b294279.txt">Table of n, a(n) for n = 1..10000</a>
%H A294279 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294279 <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-1).
%F A294279 a(n) = Sum_{i=1..floor(n/2)} i^10 + (n-i)^10.
%F A294279 From _Robert Israel_, Oct 27 2017: (Start)
%F A294279 a(2*k) = (6144*k^10-16863*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
%F A294279 a(2*k+1) = (6144*k^10+16896*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
%F A294279 G.f.: x^2*(x^20+1023*x^19+59039*x^18+1036299*x^17+9117154*x^16+48940320*x^15
%F A294279 +178348744*x^14+465661416*x^13+907378474*x^12+1340492142*x^11+1528402822*x^10
%F A294279 +1340492142*x^9+908233636*x^8+465661416*x^7+178756096*x^6+48940320*x^5
%F A294279 +9163981*x^4+1036299*x^3+60051*x^2+1023*x+2)/((x^2-1)^11*(x-1)). (End)
%F A294279 a(n) = n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584. - _Wesley Ivan Hurt_, Jul 13 2025
%F A294279 a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 55*a(n-4) + 55*a(n-5) + 165*a(n-6) - 165*a(n-7) - 330*a(n-8) + 330*a(n-9) + 462*a(n-10) - 462*a(n-11) - 462*a(n-12) + 462*a(n-13) + 330*a(n-14) - 330*a(n-15) - 165*a(n-16) + 165*a(n-17) + 55*a(n-18) - 55*a(n-19) - 11*a(n-20) + 11*a(n-21) + a(n-22) - a(n-23). - _Wesley Ivan Hurt_, Jul 13 2025
%p A294279 f:= proc(n)
%p A294279 if n::even then (1/66)*n*(6*n^10-(16863/512)*n^9+55*n^8-66*n^6+66*n^4-33*n^2+5)
%p A294279 else (1/66*(n-1))*n*(2*n-1)*(n^2-n-1)*(3*n^6-9*n^5+2*n^4+11*n^3+3*n^2-10*n-5)
%p A294279 fi end proc:
%p A294279 map(f, [$1..50]); # _Robert Israel_, Oct 27 2017
%t A294279 Table[Sum[i^10 + (n - i)^10, {i, Floor[n/2]}], {n, 30}]
%o A294279 (Magma) [n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 13 2025
%Y A294279 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), A294275 (k=8), A294276 (k=9), this sequence (k=10).
%K A294279 nonn,easy
%O A294279 1,2
%A A294279 _Wesley Ivan Hurt_, Oct 26 2017