* 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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**UPD**:

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)

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Is there a way to prove it?