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A088792
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Expansion of g.f. A(x) satisfying A(x)^2 = A(x*A(x)) / (1-x) with A(0) = 1.
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3
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1, 1, 2, 7, 38, 279, 2535, 27106, 330998, 4526827, 68402684, 1130591607, 20282026273, 392427087780, 8146462590683, 180629845193090, 4261044215912232, 106571080784076813, 2817142524095598771
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OFFSET
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0,3
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COMMENTS
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Note that if F(x)^2 = F(x*F(x)) / (1-x) with F(0) = 0, then F(x) is the g.f. of A367387. - Paul D. Hanna, Jan 08 2024
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LINKS
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FORMULA
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a(n) ~ c * n^(n + log(2)) / (exp(n) * (log(2))^n), where c = 0.9415987277996... . - Vaclav Kotesovec, Aug 08 2014
G.f. A(x) = Sum_{n>=0} a(n)*x^n and B(x) = x*A(x) satisfies the following formulas.
(1) A(x)^2 = A(x*A(x)) / (1-x).
(2) A(x) = x / ((1-x)^(1/2) * (1 - B(x))^(1/4) * (1 - B(B(x)))^(1/8) * (1 - B(B(B(x))))^(1/16) * (1 - B(B(B(B(x)))))^(1/32) * ...), an infinite product involving iterations of B(x) = x*A(x).
The iterations of B(x) = x*A(x) begin
(3.a) B(B(x)) = x*(1-x) * A(x)^3.
(3.b) B(B(B(x))) = x*(1-x)^3 * (1 - x*A(x)) * A(x)^7.
(3.c) B(B(B(B(x)))) = x*(1-x)^7 * (1 - x*A(x))^3 * (1 - x*(1-x)*A(x)^3) * A(x)^15.
(3.d) B(B(B(B(B(x))))) = x*(1-x)^15 * (1 - x*A(x))^7 * (1 - x*(1-x)*A(x)^3)^3 * (1 - x*(1-x)^3*(1-x*A(x))*A(x)^7) * A(x)^31.
The compositions of g.f. A(x) with the iterations of B(x) = x*A(x) begin
(4.a) A(B(x)) = (1-x) * A(x)^2.
(4.b) A(B(B(x))) = (1-x)^2 * (1 - x*A(x)) * A(x)^4.
(4.c) A(B(B(B(x)))) = (1-x)^4 * (1 - x*A(x))^2 * (1 - x*(1-x)*A(x)^3) * A(x)^8.
(4.d) A(B(B(B(B(x))))) = (1-x)^8 * (1 - x*A(x))^4 * (1 - x*(1-x)*A(x)^3)^2 * (1 - x*(1-x)^3*(1-x*A(x))*A(x)^7) * A(x)^16.
(End)
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EXAMPLE
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A(x) = 1 + x + 2*x^2 + 7*x^3 + 38*x^4 + 279*x^5 + 2535*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 94*x^4 + 662*x^5 + 5829*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 76*x^4 + 568*x^5 + 5167*x^6 +...
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PROG
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(PARI) {a(n)=local(A, m); if(n<1, n==0, m=1; A=1+x; for(i=1, n, A=subst(A, x, x*A+x*O(x^n))/(1-x)/A); polcoeff(A, n))}
for(n=0, 20, 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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