Let
- $T(n,k)$ be A253829 whose ordinary generating function is $$ A(x,z) = \prod\limits_{n=1}^{\infty} \frac{1-z}{1-z-xz^n}. $$
- $\nu$ be a vector of fixed length $m$ with elements $\nu_i = \delta_{1,i}$ such that for $i$ from $1$ to $m-1$, for $j$ from $i+1$ to $m$ we consecutively apply $$ \nu_j := \nu_j + x\sum\limits_{k=1}^{j-i} \nu_k. $$
I conjecture that after the whole transform we have $\nu_n = [z^{n-1}] A(x,z)$. Here is the PARI/GP program to check it numerically:
upto1(n) = {my(x = 'x, y = 'y);
Vec(prod(i=1, n, (1-x)/(1-x-y*x^i)) + x*O(x^n))}
upto2(n) = {my(y = 'y, v1);
v1 = vector(n, i, 0); v1[1] = 1;
for(i=1, n-1,
for(j=i+1, n,
v1[j] += y*sum(k=1, j-i, v1[k])));
v1}
test(n) = upto2(n+1) == upto1(n)
Is there a way to prove it?