Theory and Modern Applications
Table 1 Table of functional equations
Equation | Function B ( x ) | Composita \(\boldsymbol{B_{x}^{\Delta}(n,k)}\) | OEIS |
|---|---|---|---|
\(B(x)=1+xB^{0}(x)\) | 1 + x | \({k \choose n-k}\) |  |
\(B(x)=1+xB^{1}(x)\) | \(\frac{1}{1-x}\) | \({n-1 \choose k-1}\) | A000012 |
\(B(x)=1+xB^{2}(x)\) | \(\frac{1-\sqrt{1-4x}}{2x} \) | \(\frac{k}{n}{2n-k-1 \choose n-1}\) | A000108 |
\(B(x)=1+xB^{3}(x)\) | Â | \(\frac{k}{n}{3n-2k \choose n-k}\) | A001764 |