\[\begin{matrix} 1,1,2,0 & \mathrm{Coef.}\;\frac{1-2x}{1-3x}& A133494\\ 1,2,3,0 & \mathrm{Coef.}\;\frac{1-3x}{1-5x}& A020699\\ 1,3,4,0 & \mathrm{Coef.}\;\frac{1-4x}{1-7x}& .. \\ 1,2,4,0 & \mathrm{Coef.}\;\frac{1-4x}{1-6x}& .. \\ 1,a,b,0 & \mathrm{Coef.}\;\frac{1-bx}{1-(a+b)x} & .. \\ 1,5,-6/5,31/30,-5/186,6/31,0 & & A005021 \\ 1,2,0& 2^n-0 1,3,-2/3,2/3,0&2^n-1&A000225 \\ 2,3,-2/3,2/3,0&2^n -2 & A000918 \\ 1,5,-12/5,2/5,0&2^n-3 & A036563 \\ 4,3,-2/3,2/3,0&2^n-4 & A028399 \\ 3,11/3,-40/33,6/11,0 & 2^n-5 & A168616 \\ \end{matrix}\]
\[\begin{matrix} 1,2,1,3,1,5,1,7,1,11 & & A171448 \\ 1/1,3/1,4/3,5/3,6/5,9/5,10/9,17/9,18/17,33/17 & \mathrm{Delannoy} & A001850 \\ 1,1,3,1,3,1,3,... & \mathrm{Invert} & A007564 \\ 1,1,4,1,4,1,4,... & \mathrm{Shifts left when INVERT transform applied 4 times} & A059231 \\ 1,1,5,1,5,1,5,... & \mathrm{Shifts left whern INVERT transform applied n times} & \\ \end{matrix}\]
we see the Delannoy numbers above have fractional coefficients of the form \[C=1\\ s^{(1)}_n = \frac{2^n+1}{(1-\delta_{n,1})(2^{n-1}+1)+\delta_{n,1}}\\ s^{(2)}_n = \frac{2^n+2}{2^{n}+1}\]