cp's OEIS Frontend

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.

%I A369530 #14 Feb 19 2024 08:24:20
%S A369530 1,1,3,6,18,49,150,454,1442,4599,15016,49400,164702,553109,1873688,
%T A369530 6386159,21902331,75495005,261468180,909289327,3174239650,11118613510,
%U A369530 39067873798,137664509998,486364771006,1722453449521,6113657733615,21744596455289,77488254484727,276628979514476
%N 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.
%H A369530 Paul D. Hanna, <a href="/A369530/b369530.txt">Table of n, a(n) for n = 1..500</a>
%F A369530 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A369530 (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.
%F A369530 (2) A(x) = F(0) where F(n) = x*(x^n + A(x))/(1 - F(n+1)) for n >= 0.
%F A369530 a(n) ~ c * d^n / n^(3/2), where d = 3.756512005147339026495976161... and c = 0.25870274493294899798568836... - _Vaclav Kotesovec_, Feb 19 2024
%e A369530 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 + ...
%e A369530 where A = A(x) satisfies the continued fraction
%e A369530 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)/( ... ))))))))
%e A369530 which yields a power series expansion in x starting with
%e A369530 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) + ...
%e A369530 the coefficients of which involve the Catalan numbers (A000108) and A186505.
%e A369530 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.
%e A369530 SPECIFIC VALUES.
%e A369530 A(1/6) = 0.21744636748805217628418218576669778...
%e A369530 A(1/5) = 0.28772045526015966809965759522703662...
%e A369530 A(1/4) = 0.46334623036210313649395429181658971...
%e A369530 A(0.266) = 0.643762817198125342775865466180469...
%e A369530 A(x) at x = 1/3 diverges.
%o A369530 (PARI) \\ Set N to desired number of terms
%o A369530 {my(N = 50, A = x + x*O(x^N), Q = vector(N));
%o A369530 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]));
%o A369530 A = x*(1 + A)/(1 - Q[N]) + x*O(x^N) ); Vec(A)}
%Y A369530 Cf. A186505, A000108.
%K A369530 nonn
%O A369530 1,3
%A A369530 _Paul D. Hanna_, Feb 06 2024