# Expanding a bivariate exponential generating function

Expanding an univariate exponential generating function can be done like this:

```
def egfExpand1(f, size):
x = var('x')
return taylor(f(x), x, 0, size).power_series(SR).egf_to_ogf().list()
```

For example egfExpand1(sec, 10) returns [1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521].

But how can I expand a bivariate exponential generating function? Say

```
def f(x, y): return exp(x * y) * sec(x)
def egfExpand2(f, size):
return ...
```

The expected output is an integer triangle (i.e. a list of integer lists).

The example would return an unsigned version of A119879, which starts:

```
1
0, 1
1, 0, 1
0, 3, 0, 1
5, 0, 6, 0, 1
```

**Edit:**

Frédéric suggested the following solution, slightly rewritten here.

```
def egfExpand2(f, size):
y = polygen(QQ, "y")
x = LazyPowerSeriesRing(y.parent(), "x").gen()
return [list(f(x, y)[n] * factorial(n)) for n in range(size)]
f = lambda x, y: exp(x * y) * sec(x)
egfExpand2(f, 10)
```

The univariate case can also be written more elegantly with this method:

```
def egfExpand1(f, size: int):
x = LazyPowerSeriesRing(QQ, "x").gen()
return [f(x)[n] * factorial(n) for n in range(size)]
egfExpand1(sec, 11)
```