TITLE: De Bruijn's sequence is odd iff $n=2^m-1$: Part I
QUESTION [8 upvotes]: Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified)
$$\hat{S}(4,n)=\frac1{n+1}\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4.$$
An all-familiar fact states: the Catalan number $C_n=\frac1{n+1}\binom{2n}n$ is odd iff $n=2^m-1$. In the same tradition, I ask:

QUESTION. Is this true? $\hat{S}(4,n)$ is odd iff $n=2^m-1$ for some $m\in\mathbb{Z_{\geq0}}$.

POSTSCRIPT. This question has been extended to my other MO question.

REPLY [5 votes]: I just wished to suggest an alternative (mechanical) proof whose details can be provided (using the WZ-method) for the identity
$$\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4=\binom{2n}n\sum_{k=0}^n(-1)^k\binom{2n+k}k^2\binom{2n}{n+k}.\qquad \qquad (*)$$
The idea is: verify that both sides of $(*)$ satisfy the 2nd-order recurrence
\begin{align*}
A(n)a(n+2)- B(b)a(n+1) +C(n)a(n)=0 \end{align*}
with initial conditions $a(0)=1$ and $a(1)=14$; where
\begin{align*}
A(n)&=(2n + 3)(48n^2 + 66n + 23)(n + 2)^3 \\
B(n)&=13056n^6+ 96288n^5+ 289600n^4+ 453428n^3 + 388698n^2 + 172598n+31030 \\
C(n)&=4(n + 1)(48n^2 + 162n + 137)(2n + 1)^3. \end{align*}