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%I #11 Sep 29 2023 10:29:21
%S 1,1,2,12,130,1912,34715,743217,18255118,505070221,15532353184,
%T 525533183871,19403298048040,776437898905606,33479679336072541,
%U 1547841068340501230,76390272348430998076,4008960603544297652028,222949077434693015546579,13098226217965693342007714,810657425687536689904281842
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.
%C 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.
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.
%F (2) 1 = Sum_{n>=0} (1 + x*A(x))^(n^2) / (1 + A(x)*(1 + x*A(x))^n)^(n+1). - _Paul D. Hanna_, Dec 06 2018
%F G.f.: 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 G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.
%e G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 130*x^4 + 1912*x^5 + 34715*x^6 + 743217*x^7 + 18255118*x^8 + 505070221*x^9 + 15532353184*x^10 + ...
%e such that
%e 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 (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 for(n=0,30, print1(a(n),", "))
%Y Cf. A303927, A303928, A303923, A303056.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 03 2018
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