Let $\mathscr{B}_t(Z)$ be a generalized binomial series $$\mathscr{B}_t(z)=\sum\limits_{k=0}^{\infty}{tk+1\choose k} \dfrac{1}{tk+1}z^k.$$ The answer follows from these two formulae \begin{gather} \tag{1}\mathscr{B}_t^r(z)=\sum\limits_{k=0}^{\infty}{tk+r\choose k} \dfrac{r}{tk+r}z^k,\\ \tag{2}\log\mathscr{B}_t(z)=\sum\limits_{k=1}^{\infty}{t k\choose k}\frac{z^k}{tk} \end{gather} because $$F(k,x)=k(1-k)\log\mathscr{B}_k(-x)=k(k-2)\log\mathscr{B}_{1-k}(x).$$

You can find (1) (and some other formulae) in Graham, R. L.; Knuth, D. E. & Patashnik, O. Concrete mathematics. Addison-Wesley Publishing Company, 1994.

Formula (2) follows from calculations performed in Bizley, M. Derivation of a new formula for the number of minimal lattice paths from $(0,0)$ to $(k m, k n)$ having just $t$ contacts with the line $m y = n x$ and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuaries 80, 55-62 (1954).

The proof of (1) (see Concrete Mathematics) is also based on combinatorics of paths. Probably it can give some tips for $j>1$.

See also Donald Knuth's 20th Annual Christmas Tree Lecture: (3/2)-ary Trees for additional connections and for the history of (2).

I applied these formulae in the theory of formal groups. Can you give us some background information about your question?

Let $\mathscr{B}_t(z)$ be a generalized binomial series $$\mathscr{B}_t(z)=\sum\limits_{k=0}^{\infty}{tk+1\choose k} \dfrac{1}{tk+1}z^k.$$ The answer follows from these two formulae \begin{gather} \tag{1}\mathscr{B}_t^r(z)=\sum\limits_{k=0}^{\infty}{tk+r\choose k} \dfrac{r}{tk+r}z^k,\\ \tag{2}\log\mathscr{B}_t(z)=\sum\limits_{k=1}^{\infty}{t k\choose k}\frac{z^k}{tk} \end{gather} because $$F(k,x)=k(1-k)\log\mathscr{B}_k(-x)=k(k-1)\log\mathscr{B}_{1-k}(x).$$

You can find (1) (and some other formulae) in (see Ch.5.4) Graham, R. L.; Knuth, D. E. & Patashnik, O. Concrete mathematics. Addison-Wesley Publishing Company, 1994.

Formula (2) follows from calculations performed in Bizley, M. Derivation of a new formula for the number of minimal lattice paths from $(0,0)$ to $(k m, k n)$ having just $t$ contacts with the line $m y = n x$ and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuaries 80, 55-62 (1954).

The proof of (1) (see Concrete Mathematics) is also based on combinatorics of paths. Probably it can give some tips for $j>1$.

See also Donald Knuth's 20th Annual Christmas Tree Lecture: (3/2)-ary Trees for additional connections and for the history of (2).

I applied these formulae in the theory of formal groups. Can you give us some background information about your question?

Let $\mathscr{B}_t(Z)$ be a \textit{generalized binomial series} $$\mathscr{B}_t(z)=\sum\limits_{k=0}^{\infty}{tk+1\choose k} \dfrac{1}{tk+1}z^k.$$ The answer follows from these two formulae \begin{gather} \tag{1}\mathscr{B}_t^r(z)=\sum\limits_{k=0}^{\infty}{tk+r\choose k} \dfrac{r}{tk+r}z^k,\\ \tag{2}\log\mathscr{B}_t(z)=\sum\limits_{k=1}^{\infty}{t k\choose k}\frac{z^k}{tk} \end{gather} because $$F(k,x)=k(1-k)\log\mathscr{B}_k(-x)=k(k-2)\log\mathscr{B}_{1-k}(x).$$

You can find (1) (and some other formulae) in Graham, R. L.; Knuth, D. E. & Patashnik, O. Concrete mathematics. Addison-Wesley Publishing Company, 1994.

Formula (2) follows from calculations performed in Bizley, M. Derivation of a new formula for the number of minimal lattice paths from $(0,0)$ to $(k m, k n)$ having just $t$ contacts with the line $m y = n x$ and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuaries 80, 55-62 (1954).

The proof of (1) (see Concrete Mathematics) is also based on combinatorics of paths. Probably it can give some tips for $j>1$.

See also Donald Knuth's 20th Annual Christmas Tree Lecture: (3/2)-ary Trees for additional connections and for the history of (2).

I applied these formulae in the theory of formal groups. Can you give us some background information about your question?

Let $\mathscr{B}_t(Z)$ be a generalized binomial series $$\mathscr{B}_t(z)=\sum\limits_{k=0}^{\infty}{tk+1\choose k} \dfrac{1}{tk+1}z^k.$$ The answer follows from these two formulae \begin{gather} \tag{1}\mathscr{B}_t^r(z)=\sum\limits_{k=0}^{\infty}{tk+r\choose k} \dfrac{r}{tk+r}z^k,\\ \tag{2}\log\mathscr{B}_t(z)=\sum\limits_{k=1}^{\infty}{t k\choose k}\frac{z^k}{tk} \end{gather} because $$F(k,x)=k(1-k)\log\mathscr{B}_k(-x)=k(k-2)\log\mathscr{B}_{1-k}(x).$$

You can find (1) (and some other formulae) in Graham, R. L.; Knuth, D. E. & Patashnik, O. Concrete mathematics. Addison-Wesley Publishing Company, 1994.

Formula (2) follows from calculations performed in Bizley, M. Derivation of a new formula for the number of minimal lattice paths from $(0,0)$ to $(k m, k n)$ having just $t$ contacts with the line $m y = n x$ and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuaries 80, 55-62 (1954).

The proof of (1) (see Concrete Mathematics) is also based on combinatorics of paths. Probably it can give some tips for $j>1$.

See also Donald Knuth's 20th Annual Christmas Tree Lecture: (3/2)-ary Trees for additional connections and for the history of (2).

I applied these formulae in the theory of formal groups. Can you give us some background information about your question?