Can the Hamiltonian path problem be solved by dynamic programming in $O(2^n n)$ time?
Yes, the problem can be solved in $O^*(1.657^n)$ time using algebraic methods. See the FOCS paper of Björklund [1].



[1] Björklund, Andreas. "Determinant sums for undirected Hamiltonicity." Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on. IEEE, 2010.