Let
- $a(n)$ be A382991 whose ordinary generating function is $$ A(x) = 1 + \sum\limits_{i=1}^{\infty} \prod\limits_{j=1}^{i} \left( jx + \frac{x^2}{1-x} \right). $$
- $R(n,k)$ be an integer coefficients such that $$ R(n,k) = \begin{cases} 1 & \text{if } n = 0 \\ \displaystyle{ \binom{n+k}{k} + \sum\limits_{j=0}^{k+1} \binom{k+1}{j} R(n-1, j) } & \text{otherwise} \end{cases} $$
I conjecture that $R(n-1,0) = a(n)$ for $n>0$. Here is the PARI/GP program to check it numerically:
upto1(n) = {my(x = 'x); Vec(sum(i=1, n, prod(j=1, i, j*x + x^2/(1-x) + x*O(x^n))))}
upto2(n) = {my(A = 1, x = 'x, v1, v2, v3, v4);
v1 = vector(n, i, 1);
v2 = vector(n, i, 0); v2[1] = 1;
v3 = vector(n, i, 0); v3[1] = [1];
for(i=2, n, v3[i] = Vec(A *= (x+1)));
for(i=1, n-1, v4 = v1;
for(j=1, n-i,
v1[j] = v3[i+j][j] + sum(k=1, j+1, v3[j+1][k]*v4[k]));
v2[i+1] = v1[1]);
v2}
test(n) = upto2(n) == upto1(n)
Is there a way to prove it?