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A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

%I A232192 #15 Aug 09 2018 09:47:08
%S A232192 1,1,1,5,44,519,7590,132347,2689046,62644234,1651650774,48731341965,
%T A232192 1592908456996,57173688136781,2235773294509565,94608603077007214,
%U A232192 4306708055122614542,209823573154587335730,10892496561736261641371,600171728539156939466278
%N A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.
%H A232192 Vaclav Kotesovec, <a href="/A232192/b232192.txt">Table of n, a(n) for n = 0..175</a>
%F A232192 G.f. satisfies:
%F A232192 (1) A(x) = 1 + x*Sum_{n>=0} A(x)^(n^2) / (1 + A(x)^n)^(n+1). - _Paul D. Hanna_, Mar 31 2018
%F A232192 (2) A(x) = 1 + Series_Reversion(x/G(x))
%F A232192 (3) A(x) = 1 + x*G(A(x)-1)
%F A232192 where G(x) is the g.f. of A122400, the number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
%F A232192 a(n) ~ c * d^n * n! / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.12140554666... . - _Vaclav Kotesovec_, May 07 2014
%e A232192 G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 44*x^4 + 519*x^5 + 7590*x^6 + 132347*x^7 + 2689046*x^8 + 62644234*x^9 + 1651650774*x^10 +...
%e A232192 where
%e A232192 A(x) = 1 + x + x*(A(x)-1) + x*(A(x)^2-1)^2 + x*(A(x)^3-1)^3 + x*(A(x)^4-1)^4 + x*(A(x)^5-1)^5 + x*(A(x)^6-1)^6 + x*(A(x)^7-1)^7 +...
%e A232192 Also,
%e A232192 A(x) = 1 + x/2  +  x*A(x)/(1 + A(x))^2  +  x*A(x)^4/(1 + A(x)^2)^3  +  x*A(x)^9/(1 + A(x)^3)^4  +  x*A(x)^16/(1 + A(x)^4)^5  +  x*A(x)^25/(1 + A(x)^5)^6  + ...
%o A232192 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^m-1+x*O(x^n))^m)); polcoeff(A, n)}
%o A232192 for(n=0,20,print1(a(n),", "))
%Y A232192 Cf. A122400, A192945, A192946, A192947, A192948.
%K A232192 nonn
%O A232192 0,4
%A A232192 _Paul D. Hanna_, Nov 20 2013