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nt.quantity principle – A significance sequence and Motzkin_numbers. Modular coincidence? Answer

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nt.quantity principle – A significance sequence and Motzkin_numbers. Modular coincidence?

I’ve checked out two sequences of integers, each of which have a outstanding place in combinatorics. For instance, the primary seems in Stieltjes momentum sequences for pattern-avoiding permutations (behold web page 23)
$$ a_n = sum_ {okay = 0} ^ n frac { binom {2k} okay binom {n + 1} {okay + 1} binom {n + 2} {okay + 1}} {(n +1) ^ 2 (n + 2)}. $$
The second seems in grid path enumerations as Motzkin_Numbers (a immediate cousin of the Catalan numbers)
$$ b_n = sum_ {okay = 0} ^ { lfloor frac {n} 2 rfloor} frac { binom {n} {2k} binom {2k} okay} {okay + 1}. $$

In the module $ 2 $ Arithmetic I come throughout a seemingly (fortunate) coincidence. Let me question:

QUESTION. Is that undoubted?
$$ a_n equiv b_n mod 2. $$

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