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 A294276 #15 Jul 13 2025 11:09:38
%S A294276 0,2,513,20708,282340,2255148,12313161,52928912,186884496,576258110,
%T A294276 1574304985,3942330372,9092033028,19736886008,40357579185,78935156288,
%U A294276 147520415296,266495712282,464467582161,788155279940,1299155279940,2095793274212,3300704544313
%N A294276 Sum of the ninth powers of the parts in the partitions of n into two parts.
%H A294276 Colin Barker, <a href="/A294276/b294276.txt">Table of n, a(n) for n = 1..1000</a>
%H A294276 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294276 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).
%F A294276 a(n) = Sum_{i=1..floor(n/2)} i^9 + (n-i)^9.
%F A294276 From _Colin Barker_, Nov 21 2017: (Start)
%F A294276 G.f.: x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10).
%F A294276 a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
%F A294276 (End)
%F A294276 a(n) = -n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120. - _Wesley Ivan Hurt_, Jul 12 2025
%t A294276 Table[Sum[i^9 + (n - i)^9, {i, Floor[n/2]}], {n, 40}]
%o A294276 (PARI) concat(0, Vec(x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ _Colin Barker_, Nov 21 2017
%o A294276 (Magma) [-n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294276 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), this sequence (k=9), A294279 (k=10).
%K A294276 nonn,easy
%O A294276 1,2
%A A294276 _Wesley Ivan Hurt_, Oct 26 2017