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G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * (1 + x*A(x)^n)^n.
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%I #5 Feb 24 2018 12:51:19

%S 1,1,3,10,40,176,830,4115,21198,112559,612632,3404041,19251182,

%T 110558737,643625347,3792942385,22602071988,136073693419,827100093078,

%U 5073042015385,31385390615698,195795705705357,1231410381962373,7806727084866309,49885661271015892,321311108707030967,2086162636742801262,13655153319525415679,90125387322138673911

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

%H Paul D. Hanna, <a href="/A300043/b300043.txt">Table of n, a(n) for n = 0..100</a>

%F G.f. satisfies:

%F (1) A(x) = Sum_{n>=0} x^n * A(x)^n * (1 + x*A(x)^n)^n.

%F (2) A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n*(n+1)) / (1 - x*A(x)^(n+1))^(n+1).

%F (3) A(x) = (1/x) * Series_Reversion( x/G(x) ), where G(x) = A(x/G(x)) is the g.f. of A300041.

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 40*x^4 + 176*x^5 + 830*x^6 + 4115*x^7 + 21198*x^8 + 112559*x^9 + 612632*x^10 + 3404041*x^11 + 19251182*x^12 + ...

%e such that

%e A(x) = 1 + x*A(x)*(1+x*A(x)) + x^2*A(x)^2*(1+x*A(x)^2)^2 + x^3*A(x)^3*(1+x*A(x)^3)^3 + x^4*A(x)^4*(1+x*A(x)^4)^4 + x^5*A(x)^5*(1+x*A(x)^5)^5 + x^6*A(x)^6*(1+x*A(x)^6)^6 + ...

%e The g.f. also satisfies the series identity:

%e A(x) = 1/(1-x*A(x)) + x^2*A(x)^2/(1-x*A(x)^2)^2 + x^4*A(x)^6/(1-x*A(x)^3)^3 + x^6*A(x)^12/(1-x*A(x)^4)^4 + x^8*A(x)^20/(1-x*A(x)^5)^5 + x^10*A(x)^30/(1-x*A(x)^6)^6 + ...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m * A^m * (1+x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k*(k+1))/(1 - x*A^(k+1) +x*O(x^n))^(k+1) )); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A300041, A186998, A300042.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 24 2018