Editing Binomial coefficient (section)

=== Ordinary generating functions ===
For a fixed {{mvar|n}}, the [[ordinary generating function]] of the sequence <math>\tbinom n0,\tbinom n1,\tbinom n2,\ldots</math> is
: <math>\sum_{k=0}^\infty {n\choose k} x^k = (1+x)^n.</math>

For a fixed {{mvar|k}}, the ordinary generating function of the sequence <math>\tbinom 0k,\tbinom 1k, \tbinom 2k,\ldots,</math> is
: <math>\sum_{n=0}^\infty {n\choose k} y^n = \frac{y^k}{(1-y)^{k+1}}.</math>

The [[bivariate generating function]] of the binomial coefficients is
: <math>\sum_{n=0}^\infty \sum_{k=0}^n {n\choose k} x^k y^n = \frac{1}{1-y-xy}.</math>

A symmetric bivariate generating function of the binomial coefficients is
: <math>\sum_{n=0}^\infty \sum_{k=0}^\infty {n+k\choose k} x^k y^n = \frac{1}{1-x-y}.</math>
which is the same as the previous generating function after the substitution <math>x\to xy</math>.