A369530 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.

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

Offset: 1

Author

Paul D. Hanna, Feb 06 2024

Keywords

• nonn

Examples

			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.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A186505, A000108.

Programs

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)}

Formula

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