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A174403 Expansion of (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2).

%I A174403 #34 Aug 02 2023 07:38:11
%S A174403 1,2,7,22,76,268,977,3638,13804,53164,207342,817212,3250104,13026744,
%T A174403 52567461,213394854,870845260,3570590668,14701822370,60765209876,
%U A174403 252021314536,1048538259304,4375013741962,18302920281148,76756814078840,322618359099896,1358831330368732
%N A174403 Expansion of (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2).
%C A174403 G.f. A(x) satisfies A(x)=1+2x*A(x)+2x^2*A(x)+x^2*A(x)^2. Hankel transform is A174404.
%H A174403 Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%F A174403 G.f.: 1/(1-2x-2x^2-x^2/(1-2x-2x^2-x^2/(1-... (continued fraction).
%F A174403 Let A(x) be the g.f., then B(x)=1+x*A(x) = 1 +1*x +2*x^2 +7*x^3 +22*x^4 +... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-2*x^2) (continued fraction); more generally B(x)=C(x/(1-2*x^2)) where C(x) is the g.f. for the Catalan numbers (A000108). [_Joerg Arndt_, Mar 18 2011]
%F A174403 D-finite with recurrence: (n+2)*a(n) -2*(2*n+1)*a(n-1) +4*(1-n)*a(n-2) +4*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - _R. J. Mathar_, Sep 30 2012
%F A174403 a(n) ~ 6^(1/4) * (2 + sqrt(6))^(n+1) / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 15 2018
%p A174403 with(LREtools): with(FormalPowerSeries): # requires Maple 2022
%p A174403 ogf:= (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2):
%p A174403 req:= FindRE(ogf,x,u(n));
%p A174403 init:= [1, 2, 7, 22, 76, 268]; iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
%p A174403 rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
%p A174403 a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
%p A174403 seq(a(n),n=0..24); # _Georg Fischer_, Nov 04 2022
%p A174403 with(gfun): # Alternative with gfun alone (use gfun:-version() >= 3.91):
%p A174403 FindSeq := proc(ogf) series(ogf, x, 26): [seq(coeff(%, x, n), n = 0..22)];
%p A174403 listtorec(%, r(n))[1]; subs(n=n-nops(%)-1, %); rectoproc(%, r(n), remember) end:
%p A174403 ogf := (1-sqrt((2*x^2-1)*(2*x*(x+2)-1))-2*x*(x+1))/(2*x^2):
%p A174403 a := FindSeq(ogf): seq(a(n), n=0..28); # _Peter Luschny_, Nov 04 2022
%t A174403 nmax = 24;
%t A174403 A[_] = 1;
%t A174403 Do[A[x_] = 1 + 2*x*A[x] + 2*x^2*A[x] + x^2*A[x]^2 + O[x]^(nmax+1) // Normal, {nmax}];
%t A174403 CoefficientList[A[x], x] (* _Jean-François Alcover_, Aug 02 2023 *)
%K A174403 easy,nonn
%O A174403 0,2
%A A174403 _Paul Barry_, Mar 18 2010