A294272 Sum of the fifth powers of the parts in the partitions of n into two parts.

0, 2, 33, 308, 1300, 4668, 12201, 30032, 61776, 123950, 220825, 389652, 630708, 1018808, 1539825, 2331968, 3347776, 4826682, 6657201, 9233300, 12333300, 16578452, 21571033, 28256208, 35970000, 46106918, 57617001, 72503732, 89176276, 110446800, 133987425

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 (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), this sequence (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Programs

Magma

[(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4))/192 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Mathematica

Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2]}], {n, 50}]
Table[Total[Flatten[IntegerPartitions[n,{2}]]^5],{n,35}] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{0,2,33,308,1300,4668,12201,30032,61776,123950,220825,389652,630708},40] (* Harvey P. Dale, Jun 07 2025 *)

PARI

concat(0, Vec(x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^5 + (n-i)^5.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6).

a(n) = (1/192)*(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4)).

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.

(End)