A294288 Sum of the fourth powers of the parts in the partitions of n into two distinct parts.

0, 0, 17, 82, 354, 898, 2275, 4420, 8772, 14708, 25333, 38678, 60710, 86870, 127687, 174216, 243848, 320808, 432345, 552666, 722666, 902506, 1151403, 1410508, 1763020, 2125084, 2610621, 3103646, 3756718, 4413374, 5273999, 6131984, 7246096, 8348496, 9768353

Offset: 1

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Programs

Mathematica

Table[Sum[i^4 + (n - i)^4, {i, Floor[(n-1)/2]}], {n, 40}]
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 *)
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 *)

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

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^4 + (n-i)^4.

From Colin Barker, Nov 20 2017: (Start)

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).

a(n) = (1/480)*(n*(-16 + 160*n^2 - 15*(17 + (-1)^n)*n^3 + 96*n^4)).

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.

(End)