Charcoal, 27 bytes
Ｎθ⊞υ¹Ｗ⁼¹№υ¹⊞υ⁻θ﹪⁺θ§υ±²↨υ⁰Ｉυ Try it online! Link is to verbose version of code. Outputs the nth row. Explanation: Based on @dingledooper's formula.
Ｎθ Input n.
⊞υ¹ Start with 1 as the first term in the nth row.
Ｗ⁼¹№υ¹ Repeat until there are two 1s in the row.
⊞υ⁻θ﹪⁺θ§υ±²↨υ⁰ Calculate the next term as n-(n+b)%d where d is the previous term and b is the penultimate term. Note that on the first loop there is no penultimate term so cyclic indexing gives b a value of 1 but d=1 so the result is n anyway.
Ｉυ Output all of the terms in the nth row.
A naive approach takes 34 bytes:
Ｆ²⊞υ¹Ｆ…·²Ｎ≔ΣＥυΦ⟦ικ⟧∨ν∧λ⁼ι⁺κ§υ⊖λυＩυ Attempt This Online! Link is to verbose version of code. Outputs the nth row. Explanation:
Ｆ²⊞υ¹ Start with [1, 1] as the first row.
Ｆ…·²Ｎ Loop over the remaining rows.
≔ΣＥυΦ⟦ικ⟧∨ν∧λ⁼ι⁺κ§υ⊖λυ Insert the current row number into the list at the appropriate places.
Ｉυ Output the final list.