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A143916 G.f. A(x) satisfies: A(x) = 1+x + x^2*A(x)*A'(x).

%I A143916 #9 May 19 2016 21:41:30
%S A143916 1,1,1,3,12,62,385,2781,22848,210176,2139336,23872450,289825228,
%T A143916 3803859030,53676793157,810508456373,13041332257860,222776899815744,
%U A143916 4026846590787586,76792054455516582,1540845309830989064
%N A143916 G.f. A(x) satisfies: A(x) = 1+x + x^2*A(x)*A'(x).
%H A143916 Vaclav Kotesovec, <a href="/A143916/b143916.txt">Table of n, a(n) for n = 0..445</a>
%F A143916 a(n) ~ c * n!, where constant c = A238214 / exp(1) = 0.669014536209527303065690569951975534726... - _Vaclav Kotesovec_, Feb 21 2014
%F A143916 a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * a(k) * a(n-k-1). - _Vladimir Reshetnikov_, May 17 2016
%e A143916 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 12*x^4 + 62*x^5 + 385*x^6 +...
%e A143916 A'(x) = 1 + 2*x + 9*x^2 + 48*x^3 + 310*x^4 + 2310*x^5 + 19467*x^6 +...
%e A143916 A(x)*A'(x) = 1 + 3*x + 12*x^2 + 62*x^3 + 385*x^4 + 2781*x^5 +...
%t A143916 a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, May 17 2016 *)
%o A143916 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x+x^2*deriv(A^2/2)); polcoeff(A, n)}
%Y A143916 Cf. A143917 (variant), A238214.
%K A143916 nonn
%O A143916 0,4
%A A143916 _Paul D. Hanna_, Sep 05 2008