# extract coefficients from products and sums of descending products

I need to extract coefficients of $n$ from some powers, products and sums of descending products and their reciprocals. The first 5 terms (in $x$) of the descending product $(x)_y$ are $$ x^y \left (1 - \frac{y(y-1)}{2x} + \frac{y(y-1)(y-2)(3y-1)}{24x^2} - \frac{y^2(y - 1)^2(y - 2)(y - 3)}{48x^3} + \frac{y(y-1)(y-2)(y-3)(y-4)(15y^3-30y^2+5y+2)}{5760x^4}\right) $$ and the first 5 terms of its reciprocal are $$ x^{-y} \left(1 + \frac{y(y-1)}{2x} + \frac{y(y-1)(y+1)(3y-2)}{24x^2} + \frac{y^2(y-1)^2(y+1)(y+2)}{48x^3} + \frac{y(y-1)(y+1)(y+2)(y+3)(15y^3-15y^2-10y+8)}{5760x^4}\right) $$

I need to extract the coefficients of various powers of $n$ in expressions like

$$ \frac{(l(v-1)-1)(v-1)l(l)_t}{(n-1)_{t+1}} - \left(\frac{(l(v-1)-1)(v-1)l(l)_t}{(n-1)_{t+1}}\right)^2 + \frac{n(v-1)l(l)_{2t}(l(v-1)-1)^2(l(v-1)-2)}{(n)_{2t+3}} $$ where $l = n/v$ and $n$, $t$ and $v$ are symbolic variables.

To encapsulate these descending products and their powers I wrote

```
def desc_prod(x,y):
t0 = 1
t1 = - y*(y-1)/(2*x)
t2 = (y*(y-1)*(y-2)*(3*y-1))/(24*x^2)
t3 = -(y^2*(y-1)^2*(y-2)*(y-3))/(48*x^3)
t4 = y*(y-1)*(y-2)*(y-3)*(y-4)*(15*y^3-30*y^2 + 5*y + 2)/(5760*x^4)
return (x^y)*(t0 + t1 + t2 + t3 + t4 )
def desc_prod_recip(x,y):
t0 = 1
t1 = y*(y-1)/(2*x)
t2 = (y*(y+1)*(y-1)*(3*y-2))/(24*x^2)
t3 = (y^2*(y-1)^2*(y+1)*(y+2))/(48*x^3)
t4 = y*(y-1)*(y+1)*(y+2)*(y+3)*(15*y^3-15*y^2-10*y+8)/(5760*x^4)
return (x^(-y))*(t0 + t1 + t2 + t3 + t4)
def desc_prod_power(x,y,e):
t0 = 1
t1 = - y*(y-1)/(2*x)
t2 = (y*(y-1)*(y-2)*(3*y-1))/(24*x^2)
t3 = -(y^2*(y-1)^2*(y-2)*(y-3))/(48*x^3)
t4 = y*(y-1)*(y-2)*(y-3)*(y-4)*(15*y^3-30*y^2 + 5*y + 2)/(5760*x^4)
return (x^(y*e))*(1 + multinomial(e-1,1)*t1 + multinomial(e-2,2)*t1^2 + multinomial(e-1,1)*t2 + multinomial(e-3,3)*t1^3 + multinomial(e-2,1,1)*t1*t2 + multinomial(e-4,4)*t1^4 + multinomial(e-2,2)*t2^2 + multinomial(e-3,2,1)*t1^2*t2 + multinomial(e-2,1,1)*t1*t3 + multinomial(e-1,1)*t4 )
def desc_prod_recip_power(x,y,e):
t0 = 1
t1 = y*(y-1)/(2*x)
t2 = (y*(y+1)*(y-1)*(3*y-2))/(24*x^2)
t3 = (y^2*(y-1)^2*(y+1)*(y+2))/(48*x^3)
t4 = y*(y-1)*(y+1)*(y+2)*(y+3)*(15*y^3-15*y^2-10*y+8)/(5760*x^4)
return (x^(-y*e))*(1 + multinomial(e-1,1)*t1 + multinomial(e-2,2)*t1^2 + multinomial(e-1,1)*t2 + multinomial(e-3,3)*t1^3 + multinomial(e-2,1,1)*t1*t2 + multinomial(e-4,4)*t1^4 + multinomial(e-2,2)*t2^2 + multinomial(e-3,2,1)*t1^2*t2 + multinomial(e-2,1,1)*t1*t3 + multinomial(e-1,1)*t4 )
```

but sagemath does not correctly compute the coefficients in $n$. In an attempt to figure this out I asked a question here about computing coefficients of $n$ correctly and you can see there a very simple instance of sagemath failing to compute the coefficient. In my question I tried to give the simplest situation, where I do not have powers, products nor sums of my descnding products and their reciprocals, but just enough to showcase the behaviour of sage that is getting me stuck. In his answer @dan_fulea suggests that I need to approach my problem differently. Can anyone please give me some advice about the best way to compute the coefficients of any given power of $n$ in expressions like mine.