A294279 Sum of the tenth powers of the parts in the partitions of n into two parts.

0, 2, 1025, 61098, 1108650, 10933324, 71340451, 354864276, 1427557524, 4924107550, 14914341925, 40912232702, 102769130750, 240910097848, 529882277575, 1107606410024, 2206044295976, 4225524980826, 7792505423049, 13933571680850, 24163571680850, 40869390083652

Offset: 1

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-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), A294275 (k=8), A294276 (k=9), this sequence (k=10).

Programs

Magma

[n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584 : n in [1..50]]; // Wesley Ivan Hurt, Jul 13 2025

Maple

f:= proc(n)
if n::even then (1/66)*n*(6*n^10-(16863/512)*n^9+55*n^8-66*n^6+66*n^4-33*n^2+5)
  else (1/66*(n-1))*n*(2*n-1)*(n^2-n-1)*(3*n^6-9*n^5+2*n^4+11*n^3+3*n^2-10*n-5)
fi end proc:
map(f, [$1..50]); # Robert Israel, Oct 27 2017

Mathematica

Table[Sum[i^10 + (n - i)^10, {i, Floor[n/2]}], {n, 30}]

Formula

a(n) = Sum_{i=1..floor(n/2)} i^10 + (n-i)^10.

From Robert Israel, Oct 27 2017: (Start)

a(2*k) = (6144*k^10-16863*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.

a(2*k+1) = (6144*k^10+16896*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.

G.f.: x^2*(x^20+1023*x^19+59039*x^18+1036299*x^17+9117154*x^16+48940320*x^15

[phone]*x^14+465661416*x^13+907378474*x^12+1340492142*x^11+1528402822*x^10

[phone]*x^9+908233636*x^8+465661416*x^7+178756096*x^6+48940320*x^5

+9163981*x^4+1036299*x^3+60051*x^2+1023*x+2)/((x^2-1)^11*(x-1)). (End)

a(n) = n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584. - Wesley Ivan Hurt, Jul 13 2025

a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 55*a(n-4) + 55*a(n-5) + 165*a(n-6) - 165*a(n-7) - 330*a(n-8) + 330*a(n-9) + 462*a(n-10) - 462*a(n-11) - 462*a(n-12) + 462*a(n-13) + 330*a(n-14) - 330*a(n-15) - 165*a(n-16) + 165*a(n-17) + 55*a(n-18) - 55*a(n-19) - 11*a(n-20) + 11*a(n-21) + a(n-22) - a(n-23). - Wesley Ivan Hurt, Jul 13 2025