Simplify the following sum on binomial numbers

Question

I need help to find out any way to get simplified the following binomial coefficients sum:

$$\sum_{j=0}^{b}{\binom{b+j-1}{b-j-1} \binom{b-j+a}{a}(-2)^{b-j}}$$

No clear idea on where to begin.

Answer 1

I'm not hopeful about this. I wrote a python script to generate the first few polynomials where we hold $b$ fixed and let $a$ vary. In each case, I used sympy to interpolate a polynomial from the first $b+1$ values. It then checks that this is gives a polynomial of degree $b$ and tests that the next $b$ values are indeed given by the polynomial.

from sympy import factorial as fact
from sympy import interpolate, poly
from sympy.abc import x

def choose(n,m):
    if n<0 or m<0:
        return 0
    return fact(n)//(fact(m)*fact(n-m))

def f(a,b):
    return sum(choose(b+j-1,b-j-1)*choose(b-j+a,a)*(-2)**(b-j) for j in range(b+1)) 

for b in range(1,10): 
    p=poly(interpolate({a:f(a, b) for a in range(b+1)}, x))
    assert p.degree() == b
    assert all(p(a)==f(a,b) for a in range(b+1,2*b))
    print(p.all_coeffs())

The output is given below for the the polynomials of degree $1$ through $9$, as a list of coefficients, leading coefficient first.

[-2, -2]
[2, 4, 2]
[-4/3, -2, 4/3, 2]
[2/3, -4/3, -44/3, -80/3, -14]
[-4/15, 8/3, 24, 202/3, 1204/15, 34]
[4/45, -32/15, -190/9, -236/3, -6524/45, -656/5, -46]
[-8/315, 52/45, 556/45, 476/9, 5104/45, 5398/45, 5276/105, 2]
[2/315, -152/315, -236/45, -944/45, -956/45, 4396/45, 109772/315, 14832/35, 178]
[-4/2835, 52/315, 1576/945, 16/5, -1064/27, -1524/5, -2817608/2835, -543238/315, -483964/315, -542]  

It doesn't seem very encouraging.