In teaching my linear algebra students about Gram-Schmidt orthogonalization, I found a curious sequence of polynomials.  They are closely related to Legendre polynomials, but they also appear to be related to Catalan numbers.  (Several of the statements below are conjectural and I am not an expert on orthogonal polynomials, so please bear with me.)

Recall that if we apply the Gram-Schmidt process to the sequence $\{1,t,t^2,t^3,...\}$, where the inner product is given by $\left<f,g\right>=\int_{-1}^1 f(t)g(t)dt$, then one obtains a the Legendre polynomials.

Doing Gram-Schmidt for the first time is always a pain, and I wanted to make the problem easier to do by hand by choosing the initial basis in such a way to avoid a lot of uncomfortable fractions.  I gave my students the set $\{1,2t,6t^2,20t^3\}$ and told them to use the inner product $\left<f,g\right>=\int_{0}^1 f(t)g(t)dt$.  (This yields a sort of "shifted" version of the Legendre polynomials.)  Notice that the coefficients of these monomials are the "central" binomial coefficients $\frac{(2n)!}{n!^2}$.  If we apply Gram-Schmidt to these, then we get the polynomials $\{1,2t-1,6t^2-6t+1,20t^3-30t^2+12t-1\}$.  (In spite of my efforts, one of my students declared that, upon finding these, he could no longer feel joy.)

The new polynomials that I want to know about are obtained by writing these "shifted" Legendre polynomials as linear combinations of the initial polynomials.  Thus, in this instance, we have
$$\left[\begin{array}{c}
1 \\
t-1 \\
6t^2-6t+1 \\
20t^3-30t^2+12t-1 \\
\end{array}\right]
=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
-1 & 1 & 0 & 0 \\
1 & -3 & 1 & 0 \\
-1 & 6 & -5 & 1 \\
\end{array}\right]
\left[\begin{array}{c}
1 \\
2t \\
6t^2 \\
20t^3 \\
\end{array}\right].$$
We use the coefficient matrix here to define a new sequence by
$$\left[\begin{array}{c}
f_0 \\
f_1 \\
f_2 \\
f_3 \\
\end{array}\right]
=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
-1 & 1 & 0 & 0 \\
1 & -3 & 1 & 0 \\
-1 & 6 & -5 & 1 \\
\end{array}\right]
\left[\begin{array}{c}
1 \\
t \\
t^2 \\
t^3 \\
\end{array}\right]
=\left[\begin{array}{c}
1 \\
t-1 \\
t^2-3t+1 \\
t^3+6t^2-5t+1 \\
\end{array}\right].$$
I hope that the reader now understands how to define $f_n$ for all $n$.  Use the coefficients that are required to write the orthogonal polynomials with respect to the monomials $\frac{(2n)!}{n!^2}t^n$.

My first thought about these was that they should also be a sequence of orthogonal polynomials.  They seem to have the root-interlacing property, although the roots appear to be unbounded.  This makes me think that they are related to Laguerre polynomials.  (The roots of the Legendre polynomials are necessarily between -1 and 1.)  Also, these polynomials seem to obey the (very nice) 3-term recurrence $xf_n=f_{n-1}+2f_n+f_{n+1}$ for $n=1,2,3,4,....$

This is where the Catalan numbers seem to appear.  Recall that the $n$th Catalan number is $C_n=\frac{(2n)!}{n!(n+1)!}.$  If we had an inner product on the space of polynomials such that $\left<t^i,t^j\right>=C_{i+j}$, then applying the Gram-Schmidt process to the sequence $\{1,t,t^2,t^3,t^4,...\}$ appears to yield the sequence $f_n$.  Is there a function $g$ such that $\int_0^\infty t^n g(t)dt=C_n$?  I am guessing that such $g$ should be defined on $[0,\infty)$ because of the behavior of the roots of $f_n$.

What are these polynomials called?