A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus the number of (labeled) bicolored graphs on n vertices is $$b_n=\sum_{i+j=n}\binom{n}i2^{ij}.$$

Question. Is it true that $b_n$ is never divisible by $5$? If so, any proof?

Update. One combinatorial proof and two number-theoretic proofs are supplied.

A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus the number of (labeled) bicolored graphs on n vertices is $$b_n=\sum_{i+j=n}\binom{n}i2^{ij}.$$

Question. Is it true that $b_n$ is never divisible by $5$? If so, any proof?

Update. One combinatorial proof and two number-theoretic/algebraic proofs are supplied.

A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus the number of (labeled) bicolored graphs on n vertices is $$b_n=\sum_{i+j=n}\binom{n}i2^{ij}.$$

Question. Is it true that $b_n$ is never divisible by $5$? If so, any proof?

A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus the number of (labeled) bicolored graphs on n vertices is $$b_n=\sum_{i+j=n}\binom{n}i2^{ij}.$$

Question. Is it true that $b_n$ is never divisible by $5$? If so, any proof?

Update. One combinatorial proof and two number-theoretic proofs are supplied.