A294271 Sum of the fourth powers of the parts in the partitions of n into two parts.

0, 2, 17, 114, 354, 1060, 2275, 4932, 8772, 15958, 25333, 41270, 60710, 91672, 127687, 182408, 243848, 333930, 432345, 572666, 722666, 931788, 1151403, 1451980, 1763020, 2182206, 2610621, 3180478, 3756718, 4514624, 5273999, 6263056, 7246096, 8515538, 9768353

Offset

1,2

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/2)} i^4 + (n-i)^4.

From Colin Barker, Nov 20 2017: (Start)

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).

a(n) = (1/480)*(n*(-16 + 160*n^2 + 15*(-15 + (-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/2]}], {n, 60}]
Table[Total[Flatten[IntegerPartitions[n, {2}]]^4], {n, 40}] (* Harvey P. Dale, Mar 01 2019 *)

Prog

(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
(PARI) a(n) = sum(i=1, n\2, i^4 + (n-i)^4); \\ Michel Marcus, Nov 20 2017

Crossrefs

Sequence in context: A213785 A198158 A203247 * A203123 A198043 A037747

Adjacent sequences: A294268 A294269 A294270 * A294272 A294273 A294274

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved