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E.g.f. satisfies: A(x) = exp(x*A(((x+1)^2-1)/2)).
2

%I #9 Dec 20 2014 13:45:55

%S 1,1,3,19,185,2541,45787,1037359,28649553,942585625,36294146171,

%T 1612599520599,81729515092777,4679679856932133,300257015404355115,

%U 21436580394615666991,1692530428442960006753,146987828523665177048241

%N E.g.f. satisfies: A(x) = exp(x*A(((x+1)^2-1)/2)).

%H Alois P. Heinz, <a href="/A143633/b143633.txt">Table of n, a(n) for n = 0..100</a>

%p A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply(convert(series(exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff(A(n,2)(x), x,n)*n!: seq(a(n), n=0..21);

%t A[n_, k_] := Module[{f}, f[x_] = If[n <= 0 || k == 0, 1, A[n-1, k][((x+1)^k-1)/k]]; Normal[Series[Exp[x*f[x]], { x, 0, n+1}]] /. x -> #]&; a[n_] := Coefficient[A[n, 2][x], x, n]*n!; Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Feb 14 2014, after Maple *)

%Y Cf. 2nd column of A143632.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 27 2008