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A294300 Sum of the fifth powers of the parts in the partitions of n into two distinct parts.

%I A294300 #22 Sep 04 2024 13:18:21
%S A294300 0,0,33,244,1300,4182,12201,27984,61776,117700,220825,374100,630708,
%T A294300 985194,1539825,2266432,3347776,4708584,6657201,9033300,12333300,
%U A294300 16256350,21571033,27758544,35970000,45364332,57617001,71428084,89176276,108928050,133987425
%N A294300 Sum of the fifth powers of the parts in the partitions of n into two distinct parts.
%H A294300 Colin Barker, <a href="/A294300/b294300.txt">Table of n, a(n) for n = 1..1000</a>
%H A294300 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A294300 <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 A294300 a(n) = Sum_{i=1..floor(n/2)-((n+1) mod 2)} i^5 + (n-i)^5.
%F A294300 G.f.: -x^3*(33 +211*x +858*x^2 +1616*x^3 +2178*x^4 +1656*x^5 +858*x^6 +236*x^7 +33*x^8 +x^9) /(1+x)^6 /(x-1)^7. - _R. J. Mathar_, Nov 07 2017
%F A294300 From _Colin Barker_, Nov 21 2017: (Start)
%F A294300 a(n) = (1/192)*(n^2*(-16 + 80*n^2 - 3*(33 + (-1)^n)*n^3 + 32*n^4)).
%F A294300 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 A294300 (End)
%t A294300 Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2] - Mod[n + 1, 2]}], {n, 40}]
%t A294300 Table[Total[Flatten[Select[IntegerPartitions[n,{2}],#[[1]]!=#[[2]]&]]^5],{n,40}] (* _Harvey P. Dale_, Sep 04 2024 *)
%o A294300 (PARI) concat(vector(2), Vec(x^3*(33 + 211*x + 858*x^2 + 1616*x^3 + 2178*x^4 + 1656*x^5 + 858*x^6 + 236*x^7 + 33*x^8 + x^9) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ _Colin Barker_, Nov 21 2017
%o A294300 (PARI) a(n) = sum(i=1, (n-1)\2, i^5 + (n-i)^5); \\ _Michel Marcus_, Nov 22 2017
%Y A294300 Cf. A294286, A294287, A294288.
%K A294300 nonn,easy
%O A294300 1,3
%A A294300 _Wesley Ivan Hurt_, Oct 27 2017