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A209881 G.f. satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x))].

%I A209881 #12 Jun 04 2018 18:31:21
%S A209881 1,1,4,21,136,1030,8856,84861,894928,10291986,128165720,1718395602,
%T A209881 24686953968,378444958060,6167922926704,106525443913245,
%U A209881 1943838547593888,37375737467294362,755393226726677976,16011417246585359046,355187993770520180400,8230524179585799932820
%N A209881 G.f. satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x))].
%H A209881 Vaclav Kotesovec, <a href="/A209881/b209881.txt">Table of n, a(n) for n = 0..300</a>
%F A209881 a(n) = n*A075834(n+1) for n>=1. [corrected by _Vaclav Kotesovec_, Aug 24 2017]
%F A209881 Given g.f. A(x), the g.f. of A075834 = 1 + x/(1 - x*A(x)).
%F A209881 Forms the logarithmic derivative of A075834.
%F A209881 O.g.f. A(x) satisfies: [x^n] ( 1 + x/(1 - x*A(x)) )^(n+1) = (n+1)! for n>=0.
%F A209881 O.g.f. A(x) satisfies: [x^n] exp( n * Integral A(x) dx ) * (n + 1 - A(x)) = 0 for n > 0. - _Paul D. Hanna_, Jun 04 2018
%F A209881 a(n) ~ exp(-1) * n^2 * n!. - _Vaclav Kotesovec_, Aug 24 2017
%e A209881 G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 136*x^4 + 1030*x^5 + 8856*x^6 +...
%e A209881 The g.f. of A075834, G(x) = 1/(1 - x*A(x)), begins:
%e A209881 G(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 206*x^5 + 1476*x^6 +...
%e A209881 The logarithm of the g.f. of A075834 begins:
%e A209881 log(G(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 +...
%o A209881 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A+x*O(x^n)))); polcoeff(A, n)}
%o A209881 for(n=0,25,print1(a(n),", "))
%Y A209881 Cf. A075834.
%K A209881 nonn
%O A209881 0,3
%A A209881 _Paul D. Hanna_, Mar 14 2012