A294305 Sum of the tenth powers of the parts in the partitions of n into two distinct parts.
0, 0, 1025, 59050, 1108650, 10815226, 71340451, 352767124, 1427557524, 4904576300, 14914341925, 40791300350, 102769130750, 240345147350, 529882277575, 1105458926376, 2206044295976, 4218551412024, 7792505423049, 13913571680850, 24163571680850, 40817515234450
Offset: 1
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 (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-1).
Programs
-
Mathematica
Table[Sum[i^10 + (n - i)^10, {i, Floor[(n-1)/2]}], {n, 30}] -
PARI
a(n) = sum(i=1, (n-1)\2, i^10 + (n-i)^10); \\ Michel Marcus, Nov 05 2017
-
PARI
concat(vector(2), Vec(x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11) + O(x^40))) \\ Colin Barker, Nov 21 2017
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} i^10 + (n-i)^10.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11).
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) for n>23.
(End)