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A117651 A002415 and A052472 interlaced.

%I A117651 #32 Sep 08 2022 08:45:24
%S A117651 1,0,2,1,0,6,10,20,35,50,84,105,168,196,300,336,495,540,770,825,1144,
%T A117651 1210,1638,1716,2275,2366,3080,3185,4080,4200,5304,5440,6783,6936,
%U A117651 8550,8721,10640,10830,13090,13300,15939,16170,19228,19481,23000,23276,27300
%N A117651 A002415 and A052472 interlaced.
%H A117651 G. C. Greubel, <a href="/A117651/b117651.txt">Table of n, a(n) for n = 0..5000</a>
%H A117651 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).
%F A117651 G.f.: (1 -x -2*x^2 +3*x^3 -3*x^4 +4*x^5 +16*x^6 -16*x^7 -14*x^8 +14*x^9 +4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5). - _Colin Barker_, Mar 15 2013
%F A117651 a(n) = abs((2*n^4 +12*n^3 -2*n^2 -132*n -195 +(4*n^3 -6*n^2 -124*n -189)*(-1)^n))/384. - _Luce ETIENNE_, Jun 01 2015
%F A117651 a(n) = abs((-3*(65 +63*(-1)^n) -4*(33 +31*(-1)^n)*n -2*(1+3*(-1)^n)*n^2 +4*(3 +(-1)^n)*n^3 +2*n^4)/384). - _Colin Barker_, Jun 02 2015
%F A117651 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 11. - _Charles R Greathouse IV_, Jun 02 2015
%t A117651 f[n_]:= n*(n+1)*(n+2)*(n-3)/12; g[n_]:= n^2*(n^2 -1)/12; Table[{Abs[f[n]], g[n]}, {n, 1, 25}]//Flatten
%t A117651 LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1}, {1,0,2,1,0,6,10,20,35,50,84, 105}, 50] (* _Harvey P. Dale_, Mar 05 2016 *)
%o A117651 (PARI) my(x='x+O('x^50)); Vec((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/((1+x)^4*(1-x)^5)) \\ _G. C. Greubel_, May 19 2019
%o A117651 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x -2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/( (1+x)^4*(1-x)^5) )); // _G. C. Greubel_, May 19 2019
%o A117651 (Sage) ((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, May 19 2019
%Y A117651 Cf. A002415, A052472, A018226.
%K A117651 nonn,less,easy
%O A117651 0,3
%A A117651 _Roger L. Bagula_, Apr 11 2006