Rows in Pascal's Triangle containing Numbers in Harmonic Sequence

Theorem
There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence.

Proof
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a harmonic sequence.

Then:

This is a quadratic equation in $n$, so we can calculate its discriminant.

Notice that for each $k \ge 0$:


 * $\paren {4 k + 3}^2 - 4 \paren {4 k^2 + 8 k + 4} = - \paren {8 k + 7} < 0$

By Solution to Quadratic Equation with Real Coefficients, there is no real solution for $n$.

Therefore there is no row of Pascal's triangle which contain $3$ integers in harmonic sequence.

Also see

 * Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
 * Rows in Pascal's Triangle containing Numbers in Geometric Sequence