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We prove that for each odd integer $k \geq 7$, every graph on $n$ verticeswithout odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles oflength $k$. This generalizes the previous results on the maximum number ofpentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, andasymptotically determines the generalized Tur\'an number$\mathrm{ex}(n,C_k,C_{k-2})$ for odd $k$. In contrary to the previous resultson the pentagon case, our proof is not computer-assisted.

On the maximum number of odd cycles in graphs without smaller odd cycles