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A154825 Reversion of x*(1-2*x)/(1-3*x).

%I A154825 #27 Oct 05 2024 14:34:54
%S A154825 1,-1,-1,1,5,3,-21,-51,41,391,407,-1927,-6227,2507,49347,71109,
%T A154825 -236079,-966129,9519,7408497,13685205,-32079981,-167077221,-60639939,
%U A154825 1209248505,2761755543,-4457338681,-30629783831,-22124857219,206064020315,572040039283,-590258340811
%N A154825 Reversion of x*(1-2*x)/(1-3*x).
%H A154825 Vincenzo Librandi, <a href="/A154825/b154825.txt">Table of n, a(n) for n = 0..200</a>
%F A154825 G.f.: (1+3*x-sqrt(1-2*x+9*x^2))/(4*x). - corrected by _Vaclav Kotesovec_, Feb 08 2014
%F A154825 G.f.: 1/(1+x/(1-2x/(1+x/(1-2x/(1+x/(1-2x/(1+.... (continued fraction).
%F A154825 a(n) = Sum_{k=0..n} binomial(n+k, 2k)*A000108(k)*2^k*(-3)^(n-k).
%F A154825 From _Philippe Deléham_, Jan 17 2009: (Start)
%F A154825 a(n) = Sum_{k=0..n} A131198(n,k)*(-1)^(n-k)*2^k.
%F A154825 a(n) = Sum_{k=0..n} A090181(n,k)*(-1)^k*2^(n-k).
%F A154825 a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k)*(-3)^k.
%F A154825 a(n) = Sum_{k=0..n} A088617(n,k)*2^k*(-3)^(n-k).
%F A154825 a(n) = Sum_{k=0..n} A086810(n,k)*(-1)^k*3^(n-k).
%F A154825 a(n) = Sum_{k=0..n} A133336(n,k)*3^k*(-1)^(n-k). (End)
%F A154825 D-finite with recurrence (n+1)*a(n) = (2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - _R. J. Mathar_, Nov 15 2012
%F A154825 a(n) = (-3)^n*Hypergeometric2F1([-n, n+1], [2]; 2/3). - _G. C. Greubel_, May 24 2022
%p A154825 A154825_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A154825 for w from 1 to n do a[w] := -a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1)od;
%p A154825 convert(a, list) end: A154825_list(28); # _Peter Luschny_, May 19 2011
%t A154825 CoefficientList[Series[(1+3*x-Sqrt[1-2*x+9*x^2])/(4*x), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)
%o A154825 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+3*x-Sqrt(1-2*x+9*x^2))/(4*x) )); // _G. C. Greubel_, May 24 2022
%o A154825 (SageMath) [sum(binomial(n+k,n-k)*catalan_number(k)*2^k*(-3)^(n-k) for k in (0..n)) for n in (0..40)] # _G. C. Greubel_, May 24 2022
%Y A154825 Cf. A000108, A060693, A086810, A088617, A090181, A091593, A131198, A133336.
%K A154825 sign,easy
%O A154825 0,5
%A A154825 _Paul Barry_, Jan 15 2009