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A177753 G.f.: A(x) = exp( Sum_{n>=1} (n+1)*A177752(n)*x^n/n - x ).

%I A177753 #10 Oct 18 2017 18:13:57
%S A177753 1,1,2,11,140,3102,102713,4698780,283041208,21704073515,2064570182438,
%T A177753 238616651727324,32939304929679337,5353248306115060288,
%U A177753 1011770777921642230227,220048666117424880696401
%N A177753 G.f.: A(x) = exp( Sum_{n>=1} (n+1)*A177752(n)*x^n/n - x ).
%C A177753 Let G(x) = g.f. of A177752, then A177752 is defined by:
%C A177753 . A177752(n) = [x^n] G(x)^n/(n+1) for n>1.
%H A177753 Vaclav Kotesovec, <a href="/A177753/b177753.txt">Table of n, a(n) for n = 0..250</a>
%F A177753 G.f. satisfies: 1+x + x*A'(x)/A(x) = d/dx x^2/Series_Reversion(x*A(x)).
%F A177753 a(n) ~ c * (n!)^2 / sqrt(n), where c = 0.500612869985729164508780668394780439... - _Vaclav Kotesovec_, Oct 18 2017
%e A177753 G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 3102*x^5 +...
%e A177753 Compare the series S(x) = d/dx x^2/Series_Reversion(x*A(x)):
%e A177753 S(x) = 1 + 2*x + 3*x^2 + 28*x^3 + 515*x^4 + 14766*x^5 + 596652*x^6 +...
%e A177753 to the logarithmic derivative:
%e A177753 A'(x)/A(x) = 1 + 3*x + 28*x^2 + 515*x^3 + 14766*x^4 + 596652*x^5 +...
%e A177753 and also to the g.f. G(x) of A177752:
%e A177753 G(x) = 1 + x + x^2 + 7*x^3 + 103*x^4 + 2461*x^5 + 85236*x^6 +...
%o A177753 (PARI) {a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m));A=(1/x)*serreverse(x^2/intformal(1+x+x*deriv(A)/(A+x*O(x^n))));if(n<0,0,if(n<2,1,polcoeff((n+1)*A,n)))}
%Y A177753 Cf. A177752, A293864.
%K A177753 nonn
%O A177753 0,3
%A A177753 _Paul D. Hanna_, May 16 2010