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A152681 [x^(n+1)]Reversion[x*(1-x)/(1-3*x)].

%I A152681 #25 Mar 24 2023 15:23:51
%S A152681 1,-2,2,2,-10,6,42,-102,-82,782,-814,-3854,12454,5014,-98694,142218,
%T A152681 472158,-1932258,-19038,14816994,-27370410,-64159962,334154442,
%U A152681 -121279878,-2418497010,5523511086,8914677362,-61259567662,44249714438
%N A152681 [x^(n+1)]Reversion[x*(1-x)/(1-3*x)].
%C A152681 Hankel transform is (-2)^C(n+1,2).
%H A152681 G. C. Greubel, <a href="/A152681/b152681.txt">Table of n, a(n) for n = 0..1000</a>
%F A152681 G.f.: (1 + 3*x - sqrt(1 + 2*x + 9*x^2))/(2*x).
%F A152681 a(n) = Sum_{k=0..n} C(n+k,2k)*A000108(k)*(-3)^(n-k).
%F A152681 a(n) = 0^n - 2*Sum_{k=0..floor((n-1)/2)} C(n-1,2k)*A000108(k)(-1)^(n-k-1)*2^k.
%F A152681 a(n) = Sum_{k=0..n} A090181(n,k)*(-2)^k. - _Philippe Deléham_, Feb 02 2009
%F A152681 D-finite with recurrence (n+1)*a(n) + (2*n-1)*a(n-1) + 9*(n-2)*a(n-2) = 0. - _R. J. Mathar_, Oct 25 2012
%F A152681 a(n) = (-3)^n*hypergeometric([-n, n+1], [2], 1/3). - _Peter Luschny_, Sep 17 2014
%e A152681 G.f. = 1 - 2*x + 2*x^2 + 2*x^3 - 10*x^4 + 6*x^5 + 42*x^6 + ... - _Michael Somos_, Sep 18 2018
%p A152681 A152681_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A152681 for w from 1 to n do a[w] := -2*a[w-1]+add(a[j]*a[w-j-1],j=1..w-1) od;convert(a,list)end: A152681_list(28); # _Peter Luschny_, May 19 2011
%t A152681 CoefficientList[Series[(1+3*x-Sqrt[1+2*x+9*x^2])/(2*x), {x,0,50}], x] (* _G. C. Greubel_, Sep 14 2018 *)
%o A152681 (Sage)
%o A152681 A152681 = lambda n : (-3)^n*hypergeometric([-n, n+1], [2], 1/3)
%o A152681 [round(A152681(n).n(100)) for n in (0..20)] # _Peter Luschny_, Sep 17 2014
%o A152681 (PARI) x='x+O('x^30); Vec((1 +3*x -sqrt(1+2*x+9*x^2))/(2*x)) \\ _G. C. Greubel_, Sep 14 2018
%o A152681 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 +3*x -Sqrt(1+2*x+9*x^2))/(2*x))); // _G. C. Greubel_, Sep 14 2018
%Y A152681 Cf. A125695.
%K A152681 easy,sign
%O A152681 0,2
%A A152681 _Paul Barry_, Dec 10 2008