A294275 Sum of the eighth powers of the parts in the partitions of n into two parts.

0, 2, 257, 7074, 72354, 469540, 2142595, 7972932, 24684612, 68121958, 167731333, 383769830, 812071910, 1633567432, 3103591687, 5683259528, 9961449608, 16980253770, 27957167625, 45040730666, 70540730666, 108577948908, 163239463563, 241980430540, 351625763020

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 (1,9,-9,-36,36,84,-84,-126,126,126,-126,-84,84,36,-36,-9,9,1,-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), A294272 (k=5), A294273 (k=6), A294274 (k=7), this sequence (k=8), A294276 (k=9), A294279 (k=10).

Programs

Magma

[-n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Mathematica

Table[Sum[i^8 + (n - i)^8, {i, Floor[n/2]}], {n, 40}]

PARI

concat(0, Vec( x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^8 + (n-i)^8.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9).

a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.

(End)

a(n) = -n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040. - Wesley Ivan Hurt, Jul 12 2025