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E.g.f. satisfies A(x) = exp(x*A(x^3/3!)).
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%I #13 Nov 28 2023 14:28:21

%S 1,1,1,1,5,21,61,211,1401,8065,37241,240021,1997821,14657501,

%T 105629525,958412911,9201199281,86311594881,871038486001,

%U 9432024424585,106531641929781,1271523772132741,15583607760968941,194983864950339851

%N E.g.f. satisfies A(x) = exp(x*A(x^3/3!)).

%H Alois P. Heinz, <a href="/A143567/b143567.txt">Table of n, a(n) for n = 0..100</a>

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (3*k+1) * a(k) * a(n-1-3*k) / (6^k * k! * (n-1-3*k)!). - _Seiichi Manyama_, Nov 28 2023

%p A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-3)(x^3/6)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..29);

%t A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^3/3!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient [A[n][x], x, n]*n!; Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Feb 13 2014, after Maple *)

%Y 3rd column of A143565.

%Y Cf. A367719.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Aug 24 2008, Aug 25 2008