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A114014 Expansion of g.f. (1 + 2*x)^4/((1 + x)*(1 - 16*x^2)).

%I A114014 #22 Aug 07 2023 16:33:05
%S A114014 1,7,33,127,529,2031,8465,32495,135441,519919,2167057,8318703,
%T A114014 34672913,133099247,554766609,2129587951,8876265745,34073407215,
%U A114014 142020251921,545174515439,2272324030737,8722792247023,36357184491793
%N A114014 Expansion of g.f. (1 + 2*x)^4/((1 + x)*(1 - 16*x^2)).
%H A114014 G. C. Greubel, <a href="/A114014/b114014.txt">Table of n, a(n) for n = 0..1000</a>
%H A114014 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,16,16).
%F A114014 G.f.: (-1)*(x + 1/2)^4/((x - 1/4)*(x + 1/4)*(x + 1)).
%F A114014 From _Colin Barker_, Dec 03 2012: (Start)
%F A114014 a(n) = (5*(-4)^n - 8*(-1)^n + 243*4^n)/120 for n>1.
%F A114014 G.f.: (1 +8*x +24*x^2 +32*x^3 +16*x^4)/((1+x)*(1-4*x)*(1+4*x)). (End)
%F A114014 From _G. C. Greubel_, Jul 07 2021: (Start)
%F A114014 a(n) = 4*a(n-1) + (-1)^n*(4^(n-1) -1)/3, n>2, with a(0) = 1, a(1) = 7, and a(2) = 33.
%F A114014 E.g.f.: (1/120)(243*exp(4*x) + 5*exp(-4*x) - 8*exp(-x) - 120*(1 + x)). (End)
%t A114014 CoefficientList[Series[(1+2*x)^4/((1+x)*(1-16*x^2)), {x, 0, 40}], x]
%t A114014 a[n_]:= a[n]= If[n<2, 7^n, If[n==2, 33, 4*a[n-1] +(-1)^n*(4^(n-1) -1)/3]];
%t A114014 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Jul 07 2021 *)
%t A114014 LinearRecurrence[{-1,16,16},{1,7,33,127,529},30] (* _Harvey P. Dale_, Aug 07 2023 *)
%o A114014 (Magma) [1,7] cat [(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n): n in [2..40]]; // _G. C. Greubel_, Jul 07 2021
%o A114014 (Sage) [1,7]+[(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n) for n in (2..40)] # _G. C. Greubel_, Jul 07 2021
%Y A114014 Cf. A112627.
%K A114014 nonn,easy
%O A114014 0,2
%A A114014 _Roger L. Bagula_, Jan 31 2006
%E A114014 New name and edited by _G. C. Greubel_, Jul 07 2021