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).

1, 2, 7, 22, 76, 268, 977, 3638, 13804, 53164, 207342, 817212, 3250104, 13026744, 52567461, 213394854, 870845260, 3570590668, 14701822370, 60765209876, 252021314536, 1048538259304, 4375013741962, 18302920281148, 76756814078840, 322618359099896, 1358831330368732

Offset: 0

Author

Paul Barry, Mar 18 2010

Keywords

• easy
• nonn

Comments

G.f. A(x) satisfies A(x)=1+2x*A(x)+2x^2*A(x)+x^2*A(x)^2. Hankel transform is A174404.

Links

• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Programs

Maple

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2):
req:= FindRE(ogf,x,u(n));
init:= [1, 2, 7, 22, 76, 268]; iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..24); # Georg Fischer, Nov 04 2022
with(gfun): # Alternative with gfun alone (use gfun:-version() >= 3.91):
FindSeq := proc(ogf) series(ogf, x, 26): [seq(coeff(%, x, n), n = 0..22)];
listtorec(%, r(n))[1]; subs(n=n-nops(%)-1, %); rectoproc(%, r(n), remember) end:
ogf := (1-sqrt((2*x^2-1)*(2*x*(x+2)-1))-2*x*(x+1))/(2*x^2):
a := FindSeq(ogf): seq(a(n), n=0..28); # Peter Luschny, Nov 04 2022

Mathematica

nmax = 24;
A[_] = 1;
Do[A[x_] = 1 + 2*x*A[x] + 2*x^2*A[x] + x^2*A[x]^2 + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 02 2023 *)

Formula

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

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]

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

a(n) ~ 6^(1/4) * (2 + sqrt(6))^(n+1) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 15 2018