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An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is utterly odd iff $k$ is odd. The first few utterly odd positive numbers $1,5,7,9,13,17,21,23,25,29,31,\dots$ are listed on OEIS.

QUESTION. Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Is this true? $$a_n:=\sum_{k=0}^n\binom{n}kC_k\equiv0\mod 2 \qquad \text{iff} \qquad \text{$n$ is an utterly odd number}.$$

This is supported experimentally.

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$C_k$ is odd iff $k=2^s-1$, $s=0,1,\ldots$. $\binom{n}{2^s-1}$ is odd iff $s$ terminal binary digits of $n$ are 1's (by Kummer's theorem). Thus the result.

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