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A272483
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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.
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3
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1, 1, 1, 2, 7, 24, 76, 240, 787, 2670, 9233, 32293, 114051, 406588, 1461748, 5293301, 19287242, 70660178, 260127781, 961814451, 3570265304, 13299988867, 49705359457, 186309387918, 700228153534, 2638299418839, 9963349661693, 37705935306758, 142978684267052, 543164138444912, 2066978553423647, 7878398598991602, 30074161433351617, 114964340210315649
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OFFSET
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1,4
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COMMENTS
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The radius of convergence of g.f. A(x) is 1/4.
Specific value S = A(1/4) = 0.4102247670209601941861161503462690608763563701332... satisfies:
(1) S = 2 * A(S^2),
(2) S = 4 * A(S^2/4) / (1 - sqrt(1 - 4*S^2)).
Limit a(n)/A000108(n-1) appears to be near 0.538...
The numerical value of this limit is 0.5377373265182445036109... . - Vaclav Kotesovec, May 07 2016
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A( A(x-x^2)^2 ) = x * A(x-x^2).
(2) A(x-x^2) = Series_Reversion( A(x^2)/x ).
(3) A( A(x^2)/x - A(x^2)^2/x^2 ) = x.
a(n) ~ c * 4^n / n^(3/2), where c = 0.075846449576603052427... . - Vaclav Kotesovec, May 07 2016
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EXAMPLE
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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 +...
such that A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2.
RELATED SERIES.
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 +...
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 +...
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 +...
where A( A(x)^2 ) = C(x)*A(x).
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 +...
where A(x-x^2) = Series_Reversion( A(x^2)/x ).
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 +...
where A( A(x-x^2)^2 ) = x*A(x-x^2).
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PROG
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(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)}
for(n=1, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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