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