A154825 - OEIS

%I #27 Oct 05 2024 14:34:54

%S 1,-1,-1,1,5,3,-21,-51,41,391,407,-1927,-6227,2507,49347,71109,

%T -236079,-966129,9519,7408497,13685205,-32079981,-167077221,-60639939,

%U 1209248505,2761755543,-4457338681,-30629783831,-22124857219,206064020315,572040039283,-590258340811

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

%H Vincenzo Librandi, <a href="/A154825/b154825.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: (1+3*x-sqrt(1-2*x+9*x^2))/(4*x). - corrected by _Vaclav Kotesovec_, Feb 08 2014

%F G.f.: 1/(1+x/(1-2x/(1+x/(1-2x/(1+x/(1-2x/(1+.... (continued fraction).

%F a(n) = Sum_{k=0..n} binomial(n+k, 2k)*A000108(k)*2^k*(-3)^(n-k).

%F From _Philippe Deléham_, Jan 17 2009: (Start)

%F a(n) = Sum_{k=0..n} A131198(n,k)*(-1)^(n-k)*2^k.

%F a(n) = Sum_{k=0..n} A090181(n,k)*(-1)^k*2^(n-k).

%F a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k)*(-3)^k.

%F a(n) = Sum_{k=0..n} A088617(n,k)*2^k*(-3)^(n-k).

%F a(n) = Sum_{k=0..n} A086810(n,k)*(-1)^k*3^(n-k).

%F a(n) = Sum_{k=0..n} A133336(n,k)*3^k*(-1)^(n-k). (End)

%F 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 a(n) = (-3)^n*Hypergeometric2F1([-n, n+1], [2]; 2/3). - _G. C. Greubel_, May 24 2022

%p A154825_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p 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 convert(a, list) end: A154825_list(28); # _Peter Luschny_, May 19 2011

%t CoefficientList[Series[(1+3*x-Sqrt[1-2*x+9*x^2])/(4*x), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%o (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 (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 Cf. A000108, A060693, A086810, A088617, A090181, A091593, A131198, A133336.

%K sign,easy

%O 0,5

%A _Paul Barry_, Jan 15 2009