(* Triangle in (R^2) *)

x:= (1-s-t)x0 + s x1 + t x2
y := (1-s-t)y0 + s y1 + t y2
F[n_,m_]:=2Integrate[Integrate[
(x^n)(y^m), 
{t, 0, 1-s}], {s, 0, 1}]

F[0,0]

Expand[3F[1,0]]

Expand[3 F[0,1]]

Expand[6 F[2,0]]

Expand[12 F[1,1]]

Expand[10 F[3,0]]

(* Line Segment in (R^2) *)

x:= (1-s)x0 + s x1 
y := (1-s)y0 + s y1
F[n_,m_]:=Integrate[
(x^n)(y^m), 
{s, 0, 1}]

F[0,0]

Expand[2F[1,0]]

Expand[3F[2,0]]

Expand[6F[1,1]]

Expand[12F[2,1]]

Expand[4F[3,0]]

(* Tetrahedron in (R^3) *)

x:= (1-s-t-u)x0 + s x1 + t x2+u x3
y := (1-s-t-u)y0 + s y1 + t y2+u y3
z:=(1-s-t-u)z0 + s z1 + t z2+u z3
F[n_,m_, p_]:=6Integrate[Integrate[Integrate[
(x^n)(y^m)(z^p), 
{u, 0, 1-s-t}],{t, 0, 1-s}], {s, 0, 1}]

F[0,0,0]

Expand[4F[1,0,0]]

Expand[20F[2,0,0]]

Expand[20 F[1,1,0]]

Expand[120 F[1,1,1]]

Expand[20 F[3,0,0]]