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A303924 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.

%I A303924 #13 Dec 13 2018 23:26:07
%S A303924 1,1,2,5,15,52,204,891,4266,22092,122358,718282,4438154,28711805,
%T A303924 193700970,1358588449,9883071724,74423630202,579231718432,
%U A303924 4652864427983,38528749877802,328519744186940,2881366257269722,25969840412367362,240307819488203558,2280902112035109237,22187847195528993904,221024332987155498348
%N A303924 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.
%H A303924 Paul D. Hanna, <a href="/A303924/b303924.txt">Table of n, a(n) for n = 0..300</a>
%F A303924 G.f. A(x) satisfies:
%F A303924 (1) 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.
%F A303924 (2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+1)) / (1 + (A(x)-1)*A(x)^n)^(n+1). - _Paul D. Hanna_, Dec 11 2018
%F A303924 G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^n - F(x))^n, where F(x) is the g.f. of A303923.
%F A303924 G.f.: x/Series_Reversion( x*G(x) ) 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 A303924 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 204*x^6 + 891*x^7 + 4266*x^8 + 22092*x^9 + 122358*x^10 + 718282*x^11 + 4438154*x^12 + ...
%e A303924 such that
%e A303924 1 = 1 + (1 + x*A(x)^2 - A(x)) + (1 + x*A(x)^3 - A(x))^2 + (1 + x*A(x)^4 - A(x))^3 + (1 + x*A(x)^5 - A(x))^4 + (1 + x*A(x)^6 - A(x))^5 + ...
%o A303924 (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+1) - Ser(A))^m ) )[#A] ); A[n+1]}
%o A303924 for(n=0,30, print1(a(n),", "))
%Y A303924 Cf. A303923, A303925.
%K A303924 nonn
%O A303924 0,3
%A A303924 _Paul D. Hanna_, May 03 2018