%I #30 Jul 24 2023 04:18:59
%S 1,1,3,9,33,125,503,2081,8849,38345,168875,753401,3398177,15469493,
%T 70984559,327982529,1524644897,7125440913,33459931155,157794990633,
%U 747021246817,3548843286829,16912921740775,80836929471329,387397148131889,1861088017162457
%N Revert transform of 2*x*(1-x)-x/(1+x).
%C Sign diagram of generating sequence: +++-------------...
%C The sequences A049171 to A049189 are defined by series reversion of a sequence with rational (ordinary) generating function g(x). Solving g(x)=y for x yields algebraic equations for x, so the sequences have P-finite recurrences. - _R. J. Mathar_, Jul 24 2023
%H Vincenzo Librandi, <a href="/A049171/b049171.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n+1) = Sum_{k=0..floor(n/2)} A108759(n,k)*2^k. - _Philippe Deléham_, Dec 08 2009
%F Recurrence: 4*(n-1)*n*a(n) = 2*(n-1)*(5*n-6)*a(n-1) + 3*(16*n^2 - 67*n + 69)*a(n-2) + (25*n^2 - 169*n + 285)*a(n-3) + (n-4)*(2*n-9)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ sqrt(sqrt(3)-1)*((5+3*sqrt(3))/2)^n/(2*sqrt(6*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%t Table[Sum[Binomial[3*k,k]*Binomial[n-1+k,3*k]/(2k+1)*2^k,{k,0,Floor[(n-1)/2]}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%K nonn
%O 1,3
%A _Olivier Gérard_
%E NAME corrected by _R. J. Mathar_, Jul 23 2023