cp's OEIS Frontend

A294273 Sum of the sixth powers of the parts in the partitions of n into two parts.

%I A294273 #14 Jul 13 2025 11:08:12
%S A294273 0,2,65,858,4890,21244,67171,188916,446964,994030,1978405,3796622,
%T A294273 6735950,11680408,19092295,30745064,47260136,71929146,105409929,
%U A294273 153455810,216455810,303993492,415601835,566623708,754740700,1003708134,1307797101,1702747126
%N A294273 Sum of the sixth powers of the parts in the partitions of n into two parts.
%H A294273 Colin Barker, <a href="/A294273/b294273.txt">Table of n, a(n) for n = 1..1000</a>
%H A294273 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294273 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).
%F A294273 a(n) = Sum_{i=1..floor(n/2)} i^6 + (n-i)^6.
%F A294273 From _Colin Barker_, Nov 20 2017: (Start)
%F A294273 G.f.: x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7).
%F A294273 a(n) = (n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7).
%F A294273 a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
%F A294273 (End)
%t A294273 Table[Sum[i^6 + (n - i)^6, {i, Floor[n/2]}], {n, 50}]
%o A294273 (PARI) concat(0, Vec(x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o A294273 (Magma) [(n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7) : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294273 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), this sequence (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
%K A294273 nonn,easy
%O A294273 1,2
%A A294273 _Wesley Ivan Hurt_, Oct 26 2017