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A303928 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.

%I A303928 #9 Dec 11 2018 22:50:01
%S A303928 1,1,4,29,312,4454,78649,1644280,39580036,1076460972,32628557331,
%T A303928 1090654903233,39861817143230,1581648436369772,67718096677762406,
%U A303928 3112120229328860775,152815413664021339930,7985028281346030147672,442406826626726978612624,25906474516335623637923581,1598761621228278791567817906
%N A303928 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.
%F A303928 G.f. A(x) satisfies:
%F A303928 (1) 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.
%F A303928 (2) 1 = Sum_{n>=0} (1 + x*A(x)^3)^(n^2) / (1 + A(x)*(1 + x*A(x)^3)^n)^(n+1). - _Paul D. Hanna_, Dec 11 2018
%F A303928 G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x)^2)^n - F(x))^n, where F(x) is the g.f. of A303927.
%F A303928 G.f.: sqrt( 1/x*Series_Reversion( x/G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^n - G(x))^n, where G(x) is the g.f. of A303926.
%e A303928 G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 312*x^4 + 4454*x^5 + 78649*x^6 + 1644280*x^7 + 39580036*x^8 + 1076460972*x^9 + 32628557331*x^10 + ...
%e A303928 such that
%e A303928 1 = 1 + ((1 + x*A(x)^3) - A(x)) + ((1 + x*A(x)^3)^2 - A(x))^2 + ((1 + x*A(x)^3)^3 - A(x))^3 + ((1 + x*A(x)^3)^4 - A(x))^4 + ((1 + x*A(x)^3)^5 - A(x))^5 + ...
%o A303928 (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A)^3)^m - Ser(A))^m ) )[#A] ); A[n+1]}
%o A303928 for(n=0,30, print1(a(n),", "))
%Y A303928 Cf. A303926, A303927.
%K A303928 nonn
%O A303928 0,3
%A A303928 _Paul D. Hanna_, May 03 2018