On a A162326 and vector $\nu$ of length $n$

Question

• Let $a(n)$ be A162326. Here $$ a(n) = \frac{1}{n}(2(5n-7)a(n-1) - 9(n-2)a(n-2)), \\ a(0) = a(1) = 1. $$ Also ordinary generating function is $$ \frac{5 - \sqrt{\frac{1-9x}{1-x}}}{4}. $$
• Let $b(n)$ 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 $[\nu_i, \nu_j] = [\nu_i + 2\nu_j, 2\nu_i + \nu_j]$.

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

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

upto1(n) = my(v1); v1 = vector(n+1, i, 0); v1[1] = 1; v1[2] = 1; for(i=2, n, v1[i+1] = (1/i)*(2*(5*i-7)*v1[i] - 9*(i-2)*v1[i-1])); v1
b(n) = my(v1); v1 = vector(n, i, 1); for(i=1, n-1, for(j=i+1, n, [v1[i], v1[j]] = [v1[i] + 2*v1[j], 2*v1[i] + v1[j]])); v1[n]
upto2(n) = my(v1); v1 = concat(1, vector(n, i, b(i)))
test(n) = upto1(n) == upto2(n)

Is there a way to prove it?