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Let $2.O_{m+1}$ denote the doubled Odd graph with vertex set $X$ on a set ofcardinality $2m+1$, where $m\geq 1$. Fix a vertex $x_0\in X$. Let$\mathcal{A}:=\mathcal{A}(x_0)$ denote the centralizer algebra of thestabilizer of $x_0$ in the automorphism group of $2.O_{m+1}$, and $T:=T(x_0)$the Terwilliger algebra of $2.O_{m+1}$. In this paper, we first give a basis of$\mathcal{A}$ by considering the action of the stabilizer of $x_0$ on $X\timesX$ and determine the dimension of $\mathcal{A}$. Furthermore, we give threesubalgebras of $\mathcal{A}$ such that their direct sum is $\mathcal{A}$ asvector space. Next, for $m\geq 3$ we find all isomorphism classes ofirreducible $T$-modules to display the decomposition of $T$ in ablock-diagonalization form. Finally, we show that the two algebras$\mathcal{A}$ and $T$ coincide. This result tells us that the graph $2.O_{m+1}$may be the first example of bipartite but not $Q$-polynomialdistance-transitive graph for which the corresponding centralizer algebra andTerwilliger algebra are equal.

The Terwilliger algebra of the doubled Odd graph