* Let $f(n)$ be an arbitrary function with integer values.
* Let $a(n)$ be an integer sequence with ordinary generating function $\frac{1}{G(0,x)}$ where $G(0,x)$ is a continued fraction such that
$$
G(k,x) = 1 - \cfrac{f(k+1)x}{G(k+1, x)}.
$$
Note that
$$
G(0, x) = 1 - \cfrac{f(1)x}{1 - \cfrac{f(2)x}{1 - \cfrac{f(3)x}{1 - \cfrac{f(4)x}{\ddots}}}}.
$$
* Let $b(n)$ be an integer sequence such that
$$
b(n) = \sum\limits_{i=0}^{n-1}a(n-i-1)b(i), \\
b(0) = 1.
$$
* Let $c(n)$ be $\nu_n$ (after the whole transform) where we start with vector $\nu$ of length $n$ with elements $\nu_i = 1$ (that is, $\nu = \{1, 1, \dotsc, 1\}$) and for $i$ from $1$ to $n-1$ and for $j$ from $i+1$ to $n$ apply $[\nu_i, \nu_j] = [\nu_i + f(j-i)\nu_j, \nu_i + f(j-i)\nu_j]$.

I conjecture that
$$
b(n) = c(n).
$$

Here is the *PARI/GP* program to check it numerically:

    f(n) = n
    c(n) = my(v1); v1 = vector(n, i, 1); for(i=1, n-1, for(j=i+1, n, A = v1[i] + f(j-i)*v1[j]; v1[i] = A; v1[j] = A)); v1[n]
    upto1(n) = my(v1); v1 = vector(n, i, c(i))
    h(n,x) = my(CF = 1); for(i=1, n, CF = 1 - f(n - i + 1)*x/CF + x*O(x^n)); 1/CF
    upto2(n) = my(v1); v1 = Vec(h(n,x)); v2 = vector(n+1, i, 0); v2[1] = 1; for(i=1, n, v2[i+1] = sum(j=0, i-1, v2[j+1]*v1[i-j])); v2 = vector(n, i, v2[i+1])
    test(n) = upto1(n) == upto2(n)

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**UPD1**:

Given conjecture can be reformulated as follows:

Start with vector $\nu$ of length $n$ with elements $\nu_i = 1$ (that is, $\nu = \{1, 1, \dotsc, 1\}$) and for $i$ from $1$ to $n-1$ and for $j$ from $i+1$ to $n$ apply $\nu_{j} = \nu_{j-1} + f(j-i)\nu_{j}$.

I conjecture that after the whole transform we have vector $\nu$ with elements $\nu_i = b(i)$.

Here is the *PARI/GP* program to check it numerically:

    f(n) = n
    upto1(n) = my(v1); v1 = vector(n, i, 1); for(i=1, n-1, for(j=i+1, n, v1[j] = v1[j-1] + f(j-i)*v1[j])); v1
    h(n,x) = my(CF = 1); for(i=1, n, CF = 1 - f(n - i + 1)*x/CF + x*O(x^n)); 1/CF
    upto2(n) = my(v1); v1 = Vec(h(n,x)); v2 = vector(n+1, i, 0); v2[1] = 1; for(i=1, n, v2[i+1] = sum(j=0, i-1, v2[j+1]*v1[i-j])); v2 = vector(n, i, v2[i+1])
    test(n) = upto1(n) == upto2(n)

**UD2**:

Is also looks like that ordinary generating function for $b(n)$ is $\frac{1}{G_1(0,x)}$ where $G_1(0,x)$ is a continued fraction such that
$$
G_1(0,x)=1-\frac{x}{G(0,x)}
$$
In other words, we have
$$
G_1(k,x)=1-\frac{f(k)x}{G_1(k+1,x)}
$$
Note that
$$
G_1(0, x) = 1 - \cfrac{f(0)x}{1 - \cfrac{f(1)x}{1 - \cfrac{f(2)x}{1 - \cfrac{f(3)x}{\ddots}}}}.
$$
Here we just need to set $f(0)=1$.

I think this interpretation is simpler because it allows you to focus on the continued fraction, rather than on two different things (namely the continued fraction and the sum).

Here is the *PARI/GP* program to check it numerically:

    f(n) = if(n == 0, 1, n)
    upto1(n) = my(v1); v1 = vector(n, i, 1); for(i=1, n-1, for(j=i+1, n, v1[j] = v1[j-1] + f(j-i)*v1[j])); v1
    h(n,x) = my(CF = 1); for(i=1, n, CF = 1 - f(n - i)*x/CF + x*O(x^n)); 1/CF
    upto2(n) = my(v1); v1 = Vec(h(n,x)); v1 = vector(n, i, v1[i+1])
    test(n) = upto1(n) == upto2(n)

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Is there a way to prove it?