|
%I #23 Jan 30 2020 21:29:16
%S 1,2,4,8,24,80,288,1088,4256,17088,70016,291584,1230592,5251584,
%T 22623232,98248704,429677056,1890700288,8364824576,37186449408,
%U 166030266368,744180244480,3347321831424,15104525959168,68357598756864
%N Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).
%H Vincenzo Librandi, <a href="/A179190/b179190.txt">Table of n, a(n) for n = 0..151</a>
%F G.f.: 2 - sqrt(1 - 4*x - 4*x^2).
%F a(n) = 4*A071356(n-2), n >= 2. - _R. J. Mathar_, Jul 08 2010
%F a(n) = Sum_{k=0..floor(n/2)} (2*n - 2*k - 3)!! *2^(n-k)/(k!*(n-2k)!), n > 0. - _R. J. Mathar_, Jul 11 2011
%F a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 26 2019
%F D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1) +4*(-n+3)*a(n-2)=0. - _R. J. Mathar_, Jan 20 2020
%e The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...
%p A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # _R. J. Mathar_, Jul 11 2011
%t Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t,0,n}], n], {n, 0, 30}] (* _G. C. Greubel_, Jan 25 2019 *)
%o (Maxima) makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); // _Bruno Berselli_, Jul 04 2011
%Y Cf. A178693, A178694, A179191.
%K nonn
%O 0,2
%A _Clark Kimberling_, Jul 01 2010
|
