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%I #15 Mar 01 2019 19:11:53
%S 0,2,17,114,354,1060,2275,4932,8772,15958,25333,41270,60710,91672,
%T 127687,182408,243848,333930,432345,572666,722666,931788,1151403,
%U 1451980,1763020,2182206,2610621,3180478,3756718,4514624,5273999,6263056,7246096,8515538,9768353
%N Sum of the fourth powers of the parts in the partitions of n into two parts.
%H Colin Barker, <a href="/A294271/b294271.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).
%F a(n) = Sum_{i=1..floor(n/2)} i^4 + (n-i)^4.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5).
%F a(n) = (1/480)*(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4)).
%F a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>11.
%F (End)
%t Table[Sum[i^4 + (n - i)^4, {i, Floor[n/2]}], {n, 60}]
%t Table[Total[Flatten[IntegerPartitions[n,{2}]]^4],{n,40}] (* _Harvey P. Dale_, Mar 01 2019 *)
%o (PARI) concat(0, Vec(x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%o (PARI) a(n) = sum(i=1, n\2, i^4 + (n-i)^4); \\ _Michel Marcus_, Nov 20 2017
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Oct 26 2017
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