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A386665 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(-1/2) )^n / ( 2 - (1+x)^n * A(x)^(-1/2) )^(n+1).

%I A386665 #8 Aug 30 2025 04:43:44
%S A386665 1,1,8,90,1336,24406,530234,13410942,388841734,12762735148,
%T A386665 469004980720,19105730068460,855146084504046,41724450644602328,
%U A386665 2204075802189470532,125300401263988607716,7626356269363721248332,494723229572772238087966,34070289390944902842701094,2482276670026891882801017812
%N A386665 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(-1/2) )^n / ( 2 - (1+x)^n * A(x)^(-1/2) )^(n+1).
%C A386665 It appears that lim_{n->oo} ( a(n+1)/a(n) )/(n+1) exists and is near 4.
%H A386665 Paul D. Hanna, <a href="/A386665/b386665.txt">Table of n, a(n) for n = 0..250</a>
%F A386665 G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = A(x)^(1/2) satisfies the following formulas.
%F A386665 (1) A(x) = Sum_{n>=0} ( (1+x)^n - 1/B(x) )^n / ( 2 - (1+x)^n/B(x) )^(n+1).
%F A386665 (2) A(x) = Sum_{n>=0} ( (1+x)^n + 1/B(x) )^n / ( 2 + (1+x)^n/B(x) )^(n+1).
%F A386665 (3) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) - 1 )^n / ( 2*B(x) - (1+x)^n )^(n+1).
%F A386665 (4) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) + 1 )^n / ( 2*B(x) + (1+x)^n )^(n+1).
%e A386665 G.f.: A(x) = 1 + x + 8*x^2 + 90*x^3 + 1336*x^4 + 24406*x^5 + 530234*x^6 + 13410942*x^7 + 388841734*x^8 + 12762735148*x^9 + 469004980720*x^10 + ...
%e A386665 RELATED SERIES.
%e A386665 A(x)^(1/2) = 1 + 2*(x/4) + 62*(x/4)^2 + 2756*(x/4)^3 + 163574*(x/4)^4 + 11997852*(x/4)^5 + 1047984172*(x/4)^6 + 106571791752*(x/4)^7 + 12417003030694*(x/4)^8 + ...
%e A386665 A(x)^(-1/2) = 1 - 2*(x/4) - 58*(x/4)^2 - 2516*(x/4)^3 - 149434*(x/4)^4 - 11055996*(x/4)^5 - 976190180*(x/4)^6 - 100318703592*(x/4)^7 - 11796814729146*(x/4)^8 - ...
%o A386665 (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(-1/2) )^m / (2 - (1+x)^m*Ser(A)^(-1/2))^(m+1) ) ) ); A[n+1]}
%o A386665 for(n=0, 30, print1(a(n), ", "))
%Y A386665 Cf. A322735, A317350, A322737.
%K A386665 nonn,new
%O A386665 0,3
%A A386665 _Paul D. Hanna_, Aug 29 2025