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A218009 Binomial transform of A212831(n).

%I A218009 #41 Aug 04 2025 04:11:08
%S A218009 0,1,4,12,30,70,160,364,824,1848,4096,8976,19488,42016,90112,192448,
%T A218009 409472,868224,1835008,3866880,8126976,17039872,35651584,74447872,
%U A218009 155187200,322959360,671088640,1392513024,2885689344,5972697088,12348030976,25501351936,52613316608
%N A218009 Binomial transform of A212831(n).
%C A218009 Companion to A217988.
%C A218009 Considering a(n+1) - 2*a(n) = 1,2,4,6,10,20,44,96,200,... = b(n), is
%C A218009 b(n+3) - 2*b(n+2) = -2,-2,0,4,8,8,0,-16,-32,-32,0,... = -A009545(n+2).
%H A218009 Vincenzo Librandi, <a href="/A218009/b218009.txt">Table of n, a(n) for n = 0..1000</a>
%H A218009 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-8).
%F A218009 G.f.: x*(1-2*x+2*x^2-2*x^3+2*x^4)/((1-2*x)^2*(1-2*x+2*x^2)). - _Bruno Berselli_, Oct 18 2012
%F A218009 a(n) = 2^(n-3)*(3*n+2)+(1+i)^(n-4)+(1-i)^(n-4), where i=sqrt(-1) and n>1, with a(0)=0, a(1)=1. - _Bruno Berselli_, Oct 18 2012
%F A218009 a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -8*a(n-4).
%t A218009 a[n_] := Sum[(1/4)*Binomial[n, k]*((-(1 + (-1)^k))*(-1 + (-1)^Floor[k/2]) - (-3 + (-1)^k)*k), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 18 2012 *)
%t A218009 CoefficientList[Series[x*(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4)/((1 - 2*x)^2*(1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 15 2012 *)
%o A218009 (Magma) I:=[0, 1, 4, 12, 30, 70]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 15 2012
%Y A218009 Cf. A212831, A214282, A214283, A217988, A009545.
%K A218009 nonn,easy
%O A218009 0,3
%A A218009 _Paul Curtz_, Oct 18 2012