- Let $a(n)$ be A338193. Here generating function is $A(x)$ such that $$ A(x) = 1 + \int\frac{\left(\frac{x}{A(x)}\right)'}{\left(\frac{x}{(A(x))^2}\right)'} \, dx. $$
- Let $$ R(n, q) = \begin{cases} 1 & \textrm{if } n = 0 \\ R(n-1, 0) + nR(n-1, 1) & \textrm{if } q = 0 \\ R(n, q-1) + n(R(n-1, q) + R(n-1, q+1)) & \textrm{otherwise} \end{cases} $$
I conjecture that $$ R(n, 0) = a(n+1). $$
Here is the PARI/GP program to check it numerically:
upto1(n) = my(v1); v1 = vector(n+1, i, 1 + (i==3)); for(i=3, n, v1[i+1] = ((12*i^2 - 35*i + 24)*v1[i] - (i-1)*(2*i^2 - 5*i - 2)*v1[i-1] + (i-4)*(i-2)*(i-1)*v1[i-2])/(2*i-1)); v1
upto2(n) = my(v1); v1 = vector(n+1, i, 1); v2 = v1; for(i=1, n, v3 = v1; v1[1] = v3[1] + i*v3[2]; for(q=1, n-i, v1[q+1] = v1[q] + i*(v3[q+1] + v3[q+2])); v2[i+1] = v1[1]); v2
test(n) = upto1(n) == concat(1, upto2(n-1))
Is there a way to prove it?
