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%I #34 Apr 24 2024 16:34:11
%S 1,1,1,2,7,24,76,240,787,2670,9233,32293,114051,406588,1461748,
%T 5293301,19287242,70660178,260127781,961814451,3570265304,13299988867,
%U 49705359457,186309387918,700228153534,2638299418839,9963349661693,37705935306758,142978684267052,543164138444912,2066978553423647,7878398598991602,30074161433351617,114964340210315649
%N G.f. A(x) satisfies: A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.
%C The radius of convergence of g.f. A(x) is 1/4.
%C Specific value S = A(1/4) = 0.4102247670209601941861161503462690608763563701332... satisfies:
%C (1) S = 2 * A(S^2),
%C (2) S = 4 * A(S^2/4) / (1 - sqrt(1 - 4*S^2)).
%C Limit a(n)/A000108(n-1) appears to be near 0.538...
%C The numerical value of this limit is 0.5377373265182445036109... . - _Vaclav Kotesovec_, May 07 2016
%H Paul D. Hanna, <a href="/A272483/b272483.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A( A(x-x^2)^2 ) = x * A(x-x^2).
%F (2) A(x-x^2) = Series_Reversion( A(x^2)/x ).
%F (3) A( A(x^2)/x - A(x^2)^2/x^2 ) = x.
%F a(n) ~ c * 4^n / n^(3/2), where c = 0.075846449576603052427... . - _Vaclav Kotesovec_, May 07 2016
%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 7*x^5 + 24*x^6 + 76*x^7 + 240*x^8 + 787*x^9 + 2670*x^10 + 9233*x^11 + 32293*x^12 +...
%e such that A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2.
%e RELATED SERIES.
%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 +...+ A000108(n-1)*x^n +...
%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 19*x^6 + 66*x^7 + 218*x^8 + 708*x^9 + 2351*x^10 + 8034*x^11 + 27980*x^12 + 98548*x^13 + 350148*x^14 +...
%e A( A(x)^2 ) = x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 30*x^6 + 96*x^7 + 312*x^8 + 1030*x^9 + 3472*x^10 + 11932*x^11 + 41619*x^12 + 146828*x^13 + 522914*x^14 +...
%e where A( A(x)^2 ) = C(x)*A(x).
%e A(x-x^2) = x - x^3 + 2*x^5 - 6*x^7 + 18*x^9 - 59*x^11 + 204*x^13 - 728*x^15 + 2672*x^17 - 10022*x^19 + 38243*x^21 - 148039*x^23 + 579954*x^25 +...
%e where A(x-x^2) = Series_Reversion( A(x^2)/x ).
%e A( A(x-x^2)^2 ) = x^2 - x^4 + 2*x^6 - 6*x^8 + 18*x^10 - 59*x^12 + 204*x^14 - 728*x^16 + 2672*x^18 - 10022*x^20 + 38243*x^22 - 148039*x^24 +...
%e where A( A(x-x^2)^2 ) = x*A(x-x^2).
%o (PARI) {a(n) = my(A=x,C=x,X=x+x*O(x^n)); for(i=1,n, C = X + C^2; A = (2*A - subst(A,x,A^2)/C )); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A272484, A371708.
%K nonn
%O 1,4
%A _Paul D. Hanna_, May 03 2016
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