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A303925 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.

%I A303925 #11 Dec 13 2018 23:26:55
%S A303925 1,1,3,12,56,288,1587,9222,55957,352267,2290842,15343839,105634437,
%T A303925 746478622,5409932286,40189454704,305972524737,2387238374532,
%U A303925 19090018863000,156496468777604,1315509548959765,11341506519584442,100300906407392783,909967403153604712,8468614126450656268,80832677102193209308,791071858022525348235
%N A303925 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.
%H A303925 Paul D. Hanna, <a href="/A303925/b303925.txt">Table of n, a(n) for n = 0..300</a>
%F A303925 G.f. A(x) satisfies:
%F A303925 (1) 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.
%F A303925 (2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+2)) / (1 + (A(x)-1)*A(x)^n)^(n+1). - _Paul D. Hanna_, Dec 11 2018
%F A303925 G.f.: 1/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 A303925 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 A303923.
%e A303925 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1587*x^6 + 9222*x^7 + 55957*x^8 + 352267*x^9 + 2290842*x^10 + 15343839*x^11 + ...
%e A303925 such that
%e A303925 1 = 1 + (1 + x*A(x)^3 - A(x)) + (1 + x*A(x)^4 - A(x))^2 + (1 + x*A(x)^5 - A(x))^3 + (1 + x*A(x)^6 - A(x))^4 + (1 + x*A(x)^7 - A(x))^5 + ...
%o A303925 (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+2) - Ser(A))^m ) )[#A] ); A[n+1]}
%o A303925 for(n=0,30, print1(a(n),", "))
%Y A303925 Cf. A303923, A303924.
%K A303925 nonn
%O A303925 0,3
%A A303925 _Paul D. Hanna_, May 03 2018