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A369530
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Expansion of g.f. A(x) satisfying A(x) = x*(1 + A(x))/(1 - x*(x + A(x))/(1 - x*(x^2 + A(x))/(1 - x*(x^3 + A(x))/(1 - ...)))), a continued fraction.
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1
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1, 1, 3, 6, 18, 49, 150, 454, 1442, 4599, 15016, 49400, 164702, 553109, 1873688, 6386159, 21902331, 75495005, 261468180, 909289327, 3174239650, 11118613510, 39067873798, 137664509998, 486364771006, 1722453449521, 6113657733615, 21744596455289, 77488254484727, 276628979514476
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x*(1 + A(x))/(1 - x*(x + A(x))/(1 - x*(x^2 + A(x))/(1 - x*(x^3 + A(x))/(1 - ...)))), a continued fraction.
(2) A(x) = F(0) where F(n) = x*(x^n + A(x))/(1 - F(n+1)) for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 3.756512005147339026495976161... and c = 0.25870274493294899798568836... - Vaclav Kotesovec, Feb 19 2024
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 6*x^4 + 18*x^5 + 49*x^6 + 150*x^7 + 454*x^8 + 1442*x^9 + 4599*x^10 + 15016*x^11 + 49400*x^12 + ...
where A = A(x) satisfies the continued fraction
A(x) = x*(1 + A)/(1 - x*(x + A)/(1 - x*(x^2 + A)/(1 - x*(x^3 + A)/(1 - x*(x^4 + A)/(1 - x*(x^5 + A)/(1 - x*(x^6 + A)/(1 - x*(x^7 + A)/( ... ))))))))
which yields a power series expansion in x starting with
A(x) = x*(1 + A) + x^2*(A + A^2) + x^3*(1 + A + 2*A^2 + 2*A^3) + x^4*(3*A + 3*A^2 + 5*A^3 + 5*A^4) + x^5*(1 + 2*A + 10*A^2 + 9*A^3 + 14*A^4 + 14*A^5) + x^6*(1 + 6*A + 10*A^2 + 33*A^3 + 28*A^4 + 42*A^5 + 42*A^6) + x^7*(1 + 8*A + 28*A^2 + 41*A^3 + 110*A^4 + 90*A^5 + 132*A^6 + 132*A^7) + x^8*(2 + 11*A + 43*A^2 + 116*A^3 + 157*A^4 + 372*A^5 + 297*A^6 + 429*A^7 + 429*A^8) + ...
the coefficients of which involve the Catalan numbers (A000108) and A186505.
The limit of iterating the above power series, substituting A with A(x) upon each pass, yields an expansion of A(x) as a power series in x alone.
SPECIFIC VALUES.
A(1/6) = 0.21744636748805217628418218576669778...
A(1/5) = 0.28772045526015966809965759522703662...
A(1/4) = 0.46334623036210313649395429181658971...
A(0.266) = 0.643762817198125342775865466180469...
A(x) at x = 1/3 diverges.
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PROG
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(PARI) \\ Set N to desired number of terms
{my(N = 50, A = x + x*O(x^N), Q = vector(N));
for(i=1, N, Q[1] = x; for(k=1, N-1, m=N-k; Q[k+1] = x*(x^m + A)/(1 - Q[k]));
A = x*(1 + A)/(1 - Q[N]) + x*O(x^N) ); Vec(A)}
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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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