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Expansion of (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2).
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%I #34 Aug 02 2023 07:38:11

%S 1,2,7,22,76,268,977,3638,13804,53164,207342,817212,3250104,13026744,

%T 52567461,213394854,870845260,3570590668,14701822370,60765209876,

%U 252021314536,1048538259304,4375013741962,18302920281148,76756814078840,322618359099896,1358831330368732

%N 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 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 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 G.f.: 1/(1-2x-2x^2-x^2/(1-2x-2x^2-x^2/(1-... (continued fraction).

%F 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 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 a(n) ~ 6^(1/4) * (2 + sqrt(6))^(n+1) / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 15 2018

%p with(LREtools): with(FormalPowerSeries): # requires Maple 2022

%p ogf:= (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2):

%p req:= FindRE(ogf,x,u(n));

%p init:= [1, 2, 7, 22, 76, 268]; iseq:= seq(u(i-1)=init[i],i=1..nops(init)):

%p rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence

%p a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):

%p seq(a(n),n=0..24); # _Georg Fischer_, Nov 04 2022

%p with(gfun): # Alternative with gfun alone (use gfun:-version() >= 3.91):

%p FindSeq := proc(ogf) series(ogf, x, 26): [seq(coeff(%, x, n), n = 0..22)];

%p listtorec(%, r(n))[1]; subs(n=n-nops(%)-1, %); rectoproc(%, r(n), remember) end:

%p ogf := (1-sqrt((2*x^2-1)*(2*x*(x+2)-1))-2*x*(x+1))/(2*x^2):

%p a := FindSeq(ogf): seq(a(n), n=0..28); # _Peter Luschny_, Nov 04 2022

%t nmax = 24;

%t A[_] = 1;

%t 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 CoefficientList[A[x], x] (* _Jean-François Alcover_, Aug 02 2023 *)

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 18 2010