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

1,3

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).

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)

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 *)

Prog

(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

Crossrefs

Sequence in context: A017671 A001159 A053820 * A296401 A259142 A142059

Adjacent sequences: A294285 A294286 A294287 * A294289 A294290 A294291

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved