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Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, <a href="https://arxiv.org/abs/2105.12274">Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers</a>, arXiv:2105.12274 [math.AG], 2021.
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b[n_] := b[n] = If[n<2, n, If[OddQ[n], 0, Function[t, t*(1-t)/2][b[n/2]]] + Sum[b[i]*b[n-i], {i, 1, n/2}]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)
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G.f.: exp(Sum_{k>=1) } B(x^k)/k), where B(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + ... = G001190(x)/x - 1 and G001190 is the g.f. for the Wedderburn-Etherington numbers A001190. - N. J. A. Sloane.
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Alois P. Heinz, <a href="/A088325/b088325.txt">Table of n, a(n) for n = 0..2542</a>
b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d+1),
d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Sep 11 2017