$$\frac{tz}{1-z}$$

\begin{aligned} L(z,t) &= 1-\frac 1{ 1+\frac{tz}{1-z} } \\ &= \frac{tz}{1-(1-t)z} \\ &= t\sum\limits_{i\ge 0} (1-t)^i z^{i+1} \end{aligned}

\begin{aligned} \widehat L(z,t) &= t\sum\limits_{i\ge 0} (1-t)^i \frac{ z^{i+1} }{(i+1)!} \\ &= \frac t{1-t} \sum\limits_{i\ge 0} \frac{ (z(1-t))^{i+1} }{(i+1)!} \\ &= \frac t{1-t} ( {\rm e} ^{z(1-t)}-1) \end{aligned}

\begin{aligned} U(z,t) &= \frac1{ 1+\frac{tz}{1-z} } \left(\left(z\frac{ {\rm d} }{ {\rm d} z}\right)\frac{tz}{1-z}\right) \frac 1{ 1+\frac z{1-z} } \\ &= \frac1{ 1+\frac{tz}{1-z} } \frac{tz}{(1-z)^2} \frac 1{ 1+\frac z{1-z} } \\ &= \frac{tz}{1-(1-t)z} \\ &= L(z,t) \\ \widehat U(z,t) &= \widehat L(z,t) \end{aligned}

$$\frac1{1-\widehat L(z,t)} \widehat U(z,t) \frac1{1-\widehat L(z,1)} = \frac{t( {\rm e} ^{z(1-t)}-1)}{(1-z)(1-t {\rm e} ^{z(1-t)})}$$

\begin{aligned} & \quad \; \left([z^n] \frac{t( {\rm e} ^{z(1-t)}-1)}{(1-z)(1-t {\rm e} ^{z(1-t)})}\right) + 1 \\ &= [z^n] \left(\frac{t( {\rm e} ^{z(1-t)}-1)}{(1-z)(1-t {\rm e} ^{z(1-t)})} + \frac 1{1-z}\right) \\ &= (1-t) [z^n] \frac1{(1-z)(1-t {\rm e} ^{z(1-t)})} \end{aligned}

\begin{aligned} [z^n] \frac1{(1-z)(1-t {\rm e} ^{z(1-t)})} = (1-t)^n [u^n] \frac1{\left(1-\frac u{1-t}\right)(1-t {\rm e} ^u)} \end{aligned}

\begin{aligned} \frac1{\left(1-\frac u{1-t}\right)(1-t {\rm e} ^u)} &= (1-t)\frac1{(1-t-u)(1-t {\rm e} ^u)} \\ &= (1-t)\left(\frac A{1-t-u} + \frac B{1-t {\rm e} ^u}\right) \end{aligned}

$$\frac{ {\rm e} ^{-u} }{(1- {\rm e} ^u+u {\rm e} ^u)(1-t {\rm e} ^u)} + \frac1{(1- {\rm e} ^u+u {\rm e} ^u)(1-t-u)}$$

\begin{aligned} [u^{n+1}] \frac{P(u)}{1-t-u} &= [u^{n+1}] P(u) \sum\limits_{i\ge 0} (t+u)^i \\ &= [u^{n+1}] P(u) \sum\limits_{j\ge 0} u^j \sum\limits_{i\ge j} \binom ij t^{i-j} \\ &\equiv \sum\limits_{k=0}^{n+1} p_k \sum\limits_{i=0}^n \binom{n-k+1+i}i t^i \pmod { t^{n+1} } \\ &= \sum\limits_{i=0}^n t^i \sum\limits_{k=0}^{n+1} p_k \binom{n-k+1+i}i \end{aligned}

\begin{aligned} [u^{n+1}] \frac{Q(F)}{1-t(1+F)} &= \frac1{n+1} [u^n] \left(\frac{Q(u)}{1-t(1+u)}\right)’ \left(\frac u{G(u)}\right)^{n+1} \\ &= \frac1{n+1} [u^n] \left(\frac{tQ(u)}{(1-t(1+u))^2}+\frac{Q'(u)}{1-t(1+u)}\right) \left(\frac uG\right)^{n+1} \end{aligned}

\begin{aligned} & \quad \; [u^n] \left(\frac{tQ(u)}{(1-t(1+u))^2}+\frac{Q'(u)}{1-t(1+u)}\right) H \\ &= [u^n] \left(Q\sum\limits_{i\ge 0}(i+1) t^{i} (1+u)^i+Q’\sum\limits_{i\ge 0} t^i (1+u)^i\right) H \end{aligned}

\begin{aligned} & \quad \; [u^n t^k] \left(\frac{tQ(u)}{(1-t(1+u))^2}+\frac{Q'(u)}{1-t(1+u)}\right) H \\ &= [u^n] \left(Q\cdot k(1+u)^{k-1}+Q’\cdot (1+u)^k\right) H \\ &= k[u^n] (1+u)^{k-1} QH + [u^n] (1+u)^k Q’H \\ &= k\sum\limits_{i=0}^n \binom{k-1}i [x^{n-i}] QH + \sum\limits_{i=0}^n \binom ki [x^{n-i}] Q’H \end{aligned}