cp's OEIS Frontend

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.

A294272 Sum of the fifth powers of the parts in the partitions of n into two parts.

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%I A294272 #16 Jul 13 2025 11:07:43
%S A294272 0,2,33,308,1300,4668,12201,30032,61776,123950,220825,389652,630708,
%T A294272 1018808,1539825,2331968,3347776,4826682,6657201,9233300,12333300,
%U A294272 16578452,21571033,28256208,35970000,46106918,57617001,72503732,89176276,110446800,133987425
%N A294272 Sum of the fifth powers of the parts in the partitions of n into two parts.
%H A294272 Colin Barker, <a href="/A294272/b294272.txt">Table of n, a(n) for n = 1..1000</a>
%H A294272 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294272 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).
%F A294272 a(n) = Sum_{i=1..floor(n/2)} i^5 + (n-i)^5.
%F A294272 From _Colin Barker_, Nov 20 2017: (Start)
%F A294272 G.f.: x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6).
%F A294272 a(n) = (1/192)*(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4)).
%F A294272 a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.
%F A294272 (End)
%t A294272 Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2]}], {n, 50}]
%t A294272 Table[Total[Flatten[IntegerPartitions[n,{2}]]^5],{n,35}] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{0,2,33,308,1300,4668,12201,30032,61776,123950,220825,389652,630708},40] (* _Harvey P. Dale_, Jun 07 2025 *)
%o A294272 (PARI) concat(0, Vec(x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o A294272 (Magma) [(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4))/192 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 12 2025
%Y A294272 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), this sequence (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
%K A294272 nonn,easy
%O A294272 1,2
%A A294272 _Wesley Ivan Hurt_, Oct 26 2017