* Let $a(n)$ be [A261041][1] (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts).
* Let $b(n)$ be an integer sequence with generating function $B(x)$ such that
$$
B(x) = (1+x)\sum\limits_{i=0}^{\infty} \frac{x^i}{\prod\limits_{j=0}^{i} (1-jx-x^2)}.
$$

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

Here is the PARI/GP program to compute $b(n)$:

    upto1(n) = Vec((1+x)*sum(i=0, n, x^i/prod(j=0, i, 1-j*x-x^2)) + x*O(x^n))
    upto2(n) = my(v1, v2, v3, v4); v1 = vector(2*n-1, i, 1); v2 = vector(2*n-1, i, 0); v3 = vector(n+1, i, 0); v3[1] = 1; v3[2] = 2; for(i=1, n-1, v4 = v1; for(j=1, 2*(n-i)-1, v1[j] = j*v1[j] + v2[j] + v1[j+1]); v3[i+2] = v3[i+1] + v1[1]; v2 = v4); v3

Is there a way to prove it?


  [1]: https://oeis.org/A261041