General expanded form of (x+y+z)^k

[sludger13]

Hi,
(hope it doesn't seem so weird),
I'm looking for a general expanded form of
[itex](x+y+z)^{k}[/itex], [itex]k\in N[/itex]

[itex]k=1[/itex]:
[itex]x+y+z[/itex]

[itex]k=2[/itex]:
[itex]x^{2}+y^{2}+z^{2}+2xy+2xz+2yz[/itex]

[itex]k=3[/itex]:
[itex]x^{3}+y^{3}+z^{3}+3xy^{2}+3xz^{2}+3yz^{2}+3x^{2}y+3x^{2}z+3y^{2}z+6xyz[/itex]

[itex]k=4[/itex]:
[itex]x^{4}+y^{4}+z^{4}+4xy^{3}+4x^{3}y+4xz^{3}+4x^{3}z+4yz^{3}[/itex]
[itex]+4y^{3}z+6x^{2}y^{2}+6y^{2}z^{2}+6x^{2}z^{2}+12x^{2}yz+12xy^{2}z+12xyz^{2}[/itex]

The elements are obviously determined by combinations of their powers, which sum is always [itex]k[/itex].
I just cannot find the algorithm for element's constants.

 

[MisterX]

http://mathworld.wolfram.com/MultinomialSeries.html
http://en.wikipedia.org/wiki/Multinomial_theorem

 

[sludger13]

Thanks, I completely forgot to check out the factorials :)