%I #26 Jan 17 2026 12:15:06

%S 0,8,510,9840,87380,488280,2015538,6725600,19173960,48427560,

%T 111111110,235794768,469070940,883708280,1589311290,2745954240,

%U 4581298448,7411742280,11668193550,17927094320,26947368420,39714002328,57489010370,81870575520,114861197400

%N a(n) = Sum_{k=1..8} n^k.

%H Alois P. Heinz, <a href="/A228292/b228292.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F G.f.: -2*x*(85*x^6+1695*x^5+7134*x^4+8254*x^3+2769*x^2+219*x+4)/(x-1)^9.

%F a(1) = 8, else a(n) = (n^9-n)/(n-1).

%F a(0)=0, a(1)=8, a(2)=510, a(3)=9840, a(4)=87380, a(5)=488280, a(6)=2015538, a(7)=6725600, a(8)=19173960, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)- 84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Harvey P. Dale_, Jan 28 2014

%F a(n) = n*A053717(n). - _Bruce Nye_, Jan 16 2026

%F E.g.f.: exp(x)*x*(8 + 247*x + 1389*x^2 + 2127*x^3 + 1206*x^4 + 288*x^5 + 29*x^6 + x^7). - _Stefano Spezia_, Jan 17 2026

%p a:= n-> `if`(n=1, 8, (n^9-n)/(n-1)):

%p seq(a(n), n=0..30);

%t Table[Total[n^Range[8]],{n,0,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{0,8,510,9840,87380,488280,2015538,6725600,19173960},30] (* _Harvey P. Dale_, Jan 28 2014 *)

%Y Column k=8 of A228275.

%Y Cf. A053717.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Aug 19 2013