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%I #22 Jan 07 2018 23:37:41
%S 0,0,17,82,354,898,2275,4420,8772,14708,25333,38678,60710,86870,
%T 127687,174216,243848,320808,432345,552666,722666,902506,1151403,
%U 1410508,1763020,2125084,2610621,3103646,3756718,4413374,5273999,6131984,7246096,8348496,9768353
%N Sum of the fourth powers of the parts in the partitions of n into two distinct parts.
%H Colin Barker, <a href="/A294288/b294288.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-1)/2)} i^4 + (n-i)^4.
%F From _Colin Barker_, Nov 20 2017: (Start)
%F G.f.: x^3*(17 + 65*x + 187*x^2 + 219*x^3 + 187*x^4 + 75*x^5 + 17*x^6 + x^7) / ((1 - x)^6*(1 + x)^5).
%F a(n) = (1/480)*(n*(-16 + 160*n^2 - 15*(17 + (-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-1)/2]}], {n, 40}]
%t Rest@ CoefficientList[ Series[ x^3*(17 +65x +187x^2 +219x^3 +187x^4 +75x^5 +17x^6 +x^7)/((1 -x)^6*(1 +x)^5), {x, 0, 35}], x] (* or *)
%t LinearRecurrence[{1, 5, -5, -10, 10, 10, -10, -5, 5, 1, -1}, {0, 0, 17, 82, 354, 898, 2275, 4420, 8772, 14708, 25333}, 35] (* _Robert G. Wilson v_, Jan 07 2018 *)
%o (PARI) concat(vector(2), Vec(x^3*(17 + 65*x + 187*x^2 + 219*x^3 + 187*x^4 + 75*x^5 + 17*x^6 + x^7) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ _Colin Barker_, Nov 20 2017
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 26 2017
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