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A274772 Zero together with the partial sums of A056640.

%I A274772 #20 Jun 19 2021 20:12:05
%S A274772 0,1,6,24,66,149,292,520,860,1345,2010,2896,4046,5509,7336,9584,12312,
%T A274772 15585,19470,24040,29370,35541,42636,50744,59956,70369,82082,95200,
%U A274772 109830,126085,144080,163936,185776,209729,235926,264504,295602,329365,365940,405480,448140,494081,543466,596464,653246,713989,778872,848080,921800,1000225,1083550
%N A274772 Zero together with the partial sums of A056640.
%C A274772 I
%H A274772 Colin Barker, <a href="/A274772/b274772.txt">Table of n, a(n) for n = 0..1000</a>
%H A274772 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A274772 a(n) = (4*n^4+8*n^3+2*n^2+4*n+3*(1-(-1)^n))/24. Therefore :
%F A274772 a(2*k) = k*(k+1)*(8*k^2+1)/3, a(2*k+1) = (k+1)*(8*k^3+16*k^2+9*k+3)/3.
%F A274772 From _Colin Barker_, Nov 11 2016: (Start)
%F A274772 G.f.: x*(1 + 2*x + 5*x^2) / ((1 - x)^5 * (1 + x)).
%F A274772 a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n>5.
%F A274772 (End)
%e A274772 a(0) = 0, a(1) = 1, a(2) = 6, a(3) = 24, a(4) = 66.
%t A274772 LinearRecurrence[{4,-5,0,5,-4,1},{0,1,6,24,66,149},60] (* _Harvey P. Dale_, Jun 19 2021 *)
%o A274772 (PARI) concat(0, Vec(x*(1 + 2*x + 5*x^2) / ((1 - x)^5 * (1 + x)) + O(x^50))) \\ _Colin Barker_, Nov 11 2016
%Y A274772 Cf. A001844, A005900, A056640.
%K A274772 nonn,easy
%O A274772 0,3
%A A274772 _Luce ETIENNE_, Nov 11 2016