%N Number of valid hook configurations of 312-avoiding permutations of [n].
%C The class of a Motzkin path is the set of indices i such that the i-th non-down step is an east step. For n > 0, a(n) is the number of pairs (P,Q) of Motzkin paths such that P and Q have the same class and P lies below or is equal to Q.
%C Conjecture: This sequence is the binomial transform of A151347.
%C The Defant article gives a functional equation that defines a generating function Q(x,y,z) such that Q(x,0,0) is the ordinary generating function of this sequence.
%H Colin Defant, <a href="http://arxiv.org/abs/1904.10451">Motzkin intervals and valid hook configurations</a>, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
%H Wenjie Fang, <a href="http://arxiv.org/abs/1801.04809">A partial order on Motzkin paths</a>, arXiv preprint arXiv:1801.04809 [math.CO], 2018.
%Y Appears to be the binomial transform of A151347.
%A _Colin Defant_, Apr 28 2019