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1, 2, 6, 22, 88, 364, 1522, 6374, 26640, 110980, 460716, 1906172, 7862416, 32341144, 132707626, 543376774, 2220650656, 9060011284, 36908739316, 150159618964, 610186287376, 2476912674664, 10044874544116, 40700948789212, 164788263075808, 666716080038824

RecurrenceTable[{n (n + 1) a[n] - 4 n (3 n - 2) a[n - 1] + 4 (2 n - 3) (6 n - 5) a[n - 2] - 16 (2 n - 5) (2 n - 3) a[n - 3] == 0, a[1] == 1, a[2] == 2, a[3] == 6}, a, {n, 1, 30}] (* Bruno Berselli, Sep 14 2018 *)

n*(n + 1)*a(n) - 4*n*(3*n - 2)*a(n-1) + 4*(2*n - 3)*(6*n - 5)*a(n-2) - 16*(2*n - 5)*(2*n - 3)*a(n-3) = 0 with n > 3. - Bruno Berselli, Sep 14 2018

G.f.: c(x)-) - 1+ + x^3(*(c'(x))^2, where c(x) is the generating function of the Catalan numbers.

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Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.

Cf. A000108. Row sums of triangles A319029 and A319030.