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%I #5 Mar 23 2019 23:52:43
%S 1,1,2,3,8,17,61,223,1058,5355,29477,171068,1042319,6646486,44231512,
%T 306592233,2209107328,16514226787,127857487521,1023541162850,
%U 8459412252464,72082183806141,632429635304865,5706629531494240,52899935984638147,503266172469569926,4909051455794089278,49053813732870894847,501726197168709837165,5248631560430224087649,56117483960904047993635,612815474656922971170469
%N G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n + sqrt(A(x)))^n / (1 + x*sqrt(A(x))*(1+x)^n)^(n+1).
%H Paul D. Hanna, <a href="/A324963/b324963.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n + sqrt(A(x)))^n / (1 + x*sqrt(A(x))*(1+x)^n)^(n+1).
%F G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n - sqrt(A(x)))^n / (1 - x*sqrt(A(x))*(1+x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 17*x^5 + 61*x^6 + 223*x^7 + 1058*x^8 + 5355*x^9 + 29477*x^10 + 171068*x^11 + 1042319*x^12 + ...
%e Let B = sqrt(A(x)) then
%e A(x) = 1/(1 + x*B) + x*((1+x) + B)/(1 + x*B*(1+x))^2 + x^2*((1+x)^2 + B)^2/(1 + x*B*(1+x)^2)^3 + x^3*((1+x)^3 + B)^3/(1 + x*B*(1+x)^3)^4 + x^4*((1+x)^4 + B)^4/(1 + x*B*(1+x)^4)^5 + x^5*((1+x)^5 + B)^5/(1 + x*B*(1+x)^5)^6 + ...
%e and
%e A(x) = 1/(1 - x*B) + x*((1+x) - B)/(1 - x*B*(1+x))^2 + x^2*((1+x)^2 - B)^2/(1 - x*B*(1+x)^2)^3 + x^3*((1+x)^3 - B)^3/(1 - x*B*(1+x)^3)^4 + x^4*((1+x)^4 - B)^4/(1 - x*B*(1+x)^4)^5 + x^5*((1+x)^5 - B)^5/(1 - x*B*(1+x)^5)^6 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n,
%o A = Vec( sum(m=0,#A, x^m*((1+x)^m + Ser(A)^(1/2))^m/(1 + x*Ser(A)^(1/2)*(1+x)^m)^(m+1)) ));A[n+1]}
%o for(n=0,35,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1,n,
%o A = Vec( sum(m=0,#A, x^m*((1+x)^m - Ser(A)^(1/2))^m/(1 - x*Ser(A)^(1/2)*(1+x)^m)^(m+1)) ));A[n+1]}
%o for(n=0,35,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 23 2019
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