Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Feb 12 2018 03:50:41
%S 0,0,1025,59050,1108650,10815226,71340451,352767124,1427557524,
%T 4904576300,14914341925,40791300350,102769130750,240345147350,
%U 529882277575,1105458926376,2206044295976,4218551412024,7792505423049,13913571680850,24163571680850,40817515234450
%N Sum of the tenth powers of the parts in the partitions of n into two distinct parts.
%H Colin Barker, <a href="/A294305/b294305.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-1).
%F a(n) = Sum_{i=1..floor((n-1)/2)} i^10 + (n-i)^10.
%F From _Colin Barker_, Nov 21 2017: (Start)
%F 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).
%F 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.
%F (End)
%t Table[Sum[i^10 + (n - i)^10, {i, Floor[(n-1)/2]}], {n, 30}]
%o (PARI) a(n) = sum(i=1, (n-1)\2, i^10 + (n-i)^10); \\ _Michel Marcus_, Nov 05 2017
%o (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
%Y Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303, A294304.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 27 2017