• Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
• Let $b(n)$ be an integer sequence with generating function $B(x)$ such that $$ B(x) = \cfrac{1}{2 - \cfrac{1}{1 - \cfrac{x}{1 - \cfrac{x}{1 - \cfrac{x^2}{1 - \cfrac{x^2}{1 - \cfrac{x^3}{1 - \cfrac{x^3}{\ddots}}}}}}}} $$

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

Is there a way to prove it?

• Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
• Let $b(n)$ be an integer sequence with generating function $B(x)$ such that $$ B(x) = \cfrac{1}{2 - \cfrac{1}{1 - \cfrac{x}{1 - \cfrac{x}{1 - \cfrac{x^2}{1 - \cfrac{x^2}{1 - \cfrac{x^3}{1 - \cfrac{x^3}{\ddots}}}}}}}} $$

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

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

upto1(n) = my(CF = 1); for(i=1, n, CF = 1 - x^((n-i+2)\2)/CF + x*O(x^n)); Vec(1/(2 - 1/CF) + x*O(x^n))

Is there a way to prove it?