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Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and$k$ be integers satisfying $\lambda,k\geq 2$ and denote$R=\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$. Let $\delta,\alpha\in\mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give anexplicit representation and enumeration for all distinct $(\delta+\alphau^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formulato count the number of all these codes. As a corollary, we conclude that every$(\delta+\alpha u^2)$-constacyclic code over $R$ of length $2^kn$ is an idealgenerated by at most $2$ polynomials in the residue class ring $R[x]/\langlex^{2^kn}-(\delta+\alpha u^2)\rangle$.

Explicit representation for a class of Type 2 constacyclic codes over the ring $\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$ with even length