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A303923 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^n - A(x) )^n.

%I A303923 #16 Dec 13 2018 23:22:35
%S A303923 1,1,1,1,2,6,22,92,419,2066,10863,60459,354381,2177439,13979759,
%T A303923 93527819,650509643,4694372980,35086564926,271174745565,2164066408692,
%U A303923 17808271012127,150925549288155,1315804758238582,11787981398487995,108409978503340041,1022519935940220983,9882436548778410911,97788364370359938816
%N A303923 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^n - A(x) )^n.
%C A303923 Compare to: 1 = Sum_{n>=0} ( 1 + x*G(x)^k - G(x) )^n holds trivially for fixed k>0 when G(x) = 1 + x*G(x)^k ; this sequence explores the case when k varies with n.
%H A303923 Paul D. Hanna, <a href="/A303923/b303923.txt">Table of n, a(n) for n = 0..300</a>
%F A303923 G.f. A(x) satisfies:
%F A303923 (1) 1 = Sum_{n>=0} ( 1 + x*A(x)^n - A(x) )^n.
%F A303923 (2) 1 = Sum_{n>=0} x^n * A(x)^(n^2) / (1 + (A(x)-1)*A(x)^n)^(n+1). - _Paul D. Hanna_, Dec 11 2018
%F A303923 G.f.: x/Series_Reversion( x*F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^(n+1) - F(x))^n, where F(x) is the g.f. of A303924.
%F A303923 G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^(n+2) - G(x))^n, where G(x) is the g.f. of A303925.
%e A303923 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 22*x^6 + 92*x^7 + 419*x^8 + 2066*x^9 + 10863*x^10 + 60459*x^11 + 354381*x^12 + ...
%e A303923 such that
%e A303923 1 = 1 + (1 + x*A(x) - A(x)) + (1 + x*A(x)^2 - A(x))^2 + (1 + x*A(x)^3 - A(x))^3 + (1 + x*A(x)^4 - A(x))^4 + (1 + x*A(x)^5 - A(x))^5 + ...
%o A303923 (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)^m - Ser(A))^m ) )[#A] ); A[n+1]}
%o A303923 for(n=0,30, print1(a(n),", "))
%Y A303923 Cf. A303924, A303925, A303926.
%K A303923 nonn
%O A303923 0,5
%A A303923 _Paul D. Hanna_, May 03 2018