On a Question of Lehmer concerning the Comtet Numbers

The Comtet numbers $b(n,k)$ and $B(n,k)$ of the first and second kind arise from the powers of $(1+x)\log(1+x)$ and of its compositional inverse, respectively.

By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$.

The corresponding relation for the unsigned array was conjectured by M.

Kurkov, but not proved, in OEIS A354794.

We further derive explicit formulas, convolution identities, and differential-difference relations for the Touchard-type polynomials attached to the two families.