%I #20 Jun 13 2015 00:50:46

%S 1,6817,2135777,165588738,5498750979,102697107715,1264908663011,

%T 11373936899396,79985007371877,461856872635333,2269365182729029,

%U 9747136491367430,37362375267437415,129917413702762791

%N Sum of next n 8th powers.

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (18,-153,816,-3060,8568,-18564,31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

%F a(1)=1; a(n)=sum(i^n, {i, n(n-1)/2+1, n(n-1)/2+1+n}).

%F a(n) = (45n^17 + 780n^15 + 3990n^13 + 6900n^11 + 1205n^9 - 3240n^7 + 1584n^5 + 640n^3 - 384n)/11520.

%F G.f.: x*(x^16 +6799*x^15 +2013224*x^14 +128186937*x^13 +2839367964*x^12 +27332724427*x^11 +129026301848*x^10 +319786366637*x^9 +431174080326*x^8 +319786366637*x^7 +129026301848*x^6 +27332724427*x^5 +2839367964*x^4 +128186937*x^3 +2013224*x^2 +6799*x +1)/(x -1)^18. [_Colin Barker_, Sep 06 2012]

%e s=8; a(1) = 1^s = 1; a(2) = 2^8 + 3^8 = 6817; a(3) = 4^8 + 5^8 + 6^8 = 2135777, a(4) = 7^ + 8^8 + 9^8 + 10^8 = 165588738.

%t i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=8; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

%Y Cf. A072474 (s=2), A075664 - A075670 (s=3-10), A075671 (s=n).

%K nonn,easy

%O 1,2

%A _Zak Seidov_, Sep 24 2002

%E Formula from _Charles R Greathouse IV_, Sep 17 2009