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On a (lame for now) Sage 10.1.beta8 :

sage: var("x, y")
(x, y)
sage: taylor(sqrt(1+4*x^2+y^2),(x, 1), (y, 2), 2)
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3
sage: var("u, v")
(u, v)
sage: taylor(sqrt(1+4*x^2+y^2).subs({x:u+1, y:v+2}),(u, 0), (v, 0), 2).subs({u:x
....: -1, v:y-2})
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3

Sage insists to develop degree 1 monomials with integer (numerical) coefficients ; this has already caused much tears, wails and teeth grindings (on this site and elsewhere). Complain fiercely...

HTH,

On a (lame for now) Sage 10.1.beta8 :

sage: var("x, y")
(x, y)
sage: taylor(sqrt(1+4*x^2+y^2),(x, 1), (y, 2), 2)
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3
sage: var("u, v")
(u, v)
sage: taylor(sqrt(1+4*x^2+y^2).subs({x:u+1, y:v+2}),(u, 0), (v, 0), 2).subs({u:x
....: -1, v:y-2})
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3
sage: 1/2*(x-1)+2/3*(y-2)
1/2*x + 2/3*y - 11/6

Sage insists to develop degree 1 monomials with integer (numerical) coefficients ; this has already caused much tears, wails and teeth grindings (on this site and elsewhere). Complain fiercely...

HTH,

On a (lame for now) Sage 10.1.beta8 :

sage: var("x, y")
(x, y)
sage: taylor(sqrt(1+4*x^2+y^2),(x, 1), (y, 2), 2)
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3
sage: var("u, v")
(u, v)
sage: taylor(sqrt(1+4*x^2+y^2).subs({x:u+1, y:v+2}),(u, 0), (v, 0), 2).subs({u:x
....: -1, v:y-2})
10/27*(x - 1)^2 - 8/27*(x - 1)*(y - 2) + 5/54*(y - 2)^2 + 4/3*x + 2/3*y + 1/3
sage: 1/2*(x-1)+2/3*(y-2)
1/2*x + 2/3*y - 11/6

Sage insists to develop degree 1 monomials with integer (numerical) numerical coefficients ; this :

sage: 1/2*(x-1)+2/3*(y-2)
1/2*x + 2/3*y - 11/6

This has already caused much tears, wails and teeth grindings (on this site and elsewhere). Complain fiercely...

EDIT : Same results on Sagecell...

HTH,