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%I #30 Sep 18 2017 12:06:39
%S 177,488,1159,2460,4781,8656,14787,24068,37609,56760,83135,118636,
%T 165477,226208,303739,401364,522785,672136,854007,1073468,1336093,
%U 1647984,2015795,2446756,2948697,3530072,4199983,4968204,5845205,6842176,7971051,9244532,10676113
%N One-fifth of the rolling arithmetic mean of the fifth powers of the natural numbers taken five at a time.
%C This method can be generalized. Replacing all the fives by any odd positive integer m, and taking m at a time, also gives an integer sequence.
%C If m is 3 then A006527 (from term 3) and A167875 (from term 2) are retrieved.
%H Colin Barker, <a href="/A292185/b292185.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = ((n^5 + (n+1)^5 + (n+2)^5 + (n+3)^5 + (n+4)^5) /5) /5.
%F From _Colin Barker_, Sep 18 2017: (Start)
%F G.f.: x*(177 - 574*x + 886*x^2 - 714*x^3 + 301*x^4 - 52*x^5) / (1 - x)^6.
%F a(n) = 6*a(n-1)- 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
%F (End)
%e a(1) = (1^5 + 2^5 + 3^5 + 4^5 +5^5)/25 = (1+32+243+1024+3125)/25 = 4425/25 = 177.
%e a(2) = (2^5 + 3^5 + 4^5 + 5^5 +6^5 )/25 = (32+243+1024+3125+7776)/25 = 12200/25 = 488.
%o (J) (m(+/ % #) \ (1+i. 44)^(x: m))%m [m=.5 NB. See http://www.jsoftware.com
%o (PARI) Vec(x*(177 - 574*x + 886*x^2 - 714*x^3 + 301*x^4 - 52*x^5) / (1 - x)^6 + O(x^30)) \\ _Colin Barker_, Sep 18 2017
%Y Cf. A006527, A167875.
%K nonn,easy
%O 1,1
%A _Robert G. Burns_, Sep 12 2017
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