Problem related to transforming polynomial


Given $a_1,a_2,…,a_m$ positive integer. Denominator $d$ is smallest positive integer for $b_l$ integer coefficient.

$$sum_{k=1}^mbinom{n}k a_k= frac{1}dsum_{l=1}^mb_ln^l$$

Now consider $n=dt+r$ where $d>rge 0$.

can it be shown that, above equation transform as

$$frac{1}dsum_{l=1}^mb_l(dt+r)^l=sum_{u=0}^{m-1}left(frac{x_ut+y_u}{z_u}right)(dt+r)^u$$

With $x_u,y_u$ and $z_u$ integers.

Example

Let $a_1=1,a_2=1,a_3=3$

then $sum_{k=1}^3binom{n}k a_k=(n^3-2n^2+3n)/2$ here $d=2$

Case$(1)$ $n=2t$,

$frac{n^3-2n^2+3n}2=(t-1)n^2+n+t$

Case$(2)$ $n=2t+1$,

$frac{n^3-2n^2+3n}2=(t-1)n^2+(t+2)n$

Thank you.