Cardinality formula conjecture for looms

Let $\mathfrak{L}_{n,m}$ be the family of looms and write $H_{n,m}=\lvert\mathfrak{L}_{n,m}\rvert$. Let $T_{m,k}$ be defined by

$$T_{m,k}=(2k+1)T_{m-1,k}+2kT_{m-1,k-1},$$

with $T_{0,0}=1$ and $T_{0,k}=0$ for every $k\geq 1$. Cardinality formula conjecture. The cardinality satisfies

$$H_{n,m}=\sum_{k=0}^m\sum_{i=0}^k(-1)^{m-i}\binom{k}{i}(2i+1)^m(2k+1)^n =\sum_{k=0}^m(-1)^{m-k}T_{m,k}(2k+1)^n.$$

The claim gives an explicit enumeration formula and an equivalent recurrence-based expression for the cardinalities of looms; the supplied text gives no evidence of resolution.