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A143566 E.g.f. satisfies A(x) = exp(x*A(x^2/2!)).

%I A143566 #20 Nov 28 2023 14:28:09
%S A143566 1,1,1,4,13,46,241,1471,9409,67348,564841,4771801,45459481,463867834,
%T A143566 5060656693,58878140686,730612429681,9556314730456,131627520296929,
%U A143566 1912237000523623,29032781640572881,462811831018070206,7687624300327129621,133275225843052767244
%N A143566 E.g.f. satisfies A(x) = exp(x*A(x^2/2!)).
%H A143566 Alois P. Heinz, <a href="/A143566/b143566.txt">Table of n, a(n) for n = 0..480</a>
%F A143566 a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (2*k+1) * a(k) * a(n-1-2*k) / (2^k * k! * (n-1-2*k)!). - _Seiichi Manyama_, Nov 28 2023
%p A143566 A:= proc(n) option remember; if n<=0 then 1 else unapply(convert(
%p A143566        series(exp(x*A(n-2)(x^2/2)), x,n+1), polynom),x) fi
%p A143566     end:
%p A143566 a:= n-> coeff(A(n)(x), x,n)*n!:
%p A143566 seq(a(n), n=0..28);
%t A143566 A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^2/2]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Feb 13 2014, after Maple *)
%Y A143566 2nd column of A143565.
%Y A143566 Cf. A138292.
%K A143566 nonn
%O A143566 0,4
%A A143566 _Alois P. Heinz_, Aug 24 2008