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A106335 Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.

%I A106335 #13 Oct 15 2023 19:07:11
%S A106335 3,2,2,6,2,7,6,3,2,6,9,2,1,9,1,1,3,3,0,9,6,9,8,7,1,3,8,6,7,3,9,8,3,0,
%T A106335 2,3,3,2,2,9,0,4,2,4,3,7,4,6,7,1,7,4,5,2,1,6,0,5,6,2,0,9,1,2,4,5,5,4,
%U A106335 8,6,2,6,7,4,1,1,1,5,0,6,4,9,7,4,7,1,2,3,7,3,9,9,1,2,2,1,4,7,8,5,3,7,1,9,0
%N A106335 Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.
%C A106335 The g.f. of A106336 equals (1/x)*Series_Reversion( x*eta(x)/eta(x^2)^2 ).
%C A106335 This constant is very close to 2^(3/2) / (3*sqrt(e*Pi)) = 0.3226276326921911330637735905807475397715626276499133673167401123748... - _Vaclav Kotesovec_, Aug 02 2017
%F A106335 Constant equals the ratio x/F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
%e A106335 x/F(x)=0.322627632692191133096987138673983023322904243746717452160562...
%e A106335 where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
%e A106335 so F(x) = 1.9873697211846841452692897833444126... (A106334)
%e A106335 at x = 0.6411803884299545796456448886283011... (A106333).
%t A106335 digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); x0/f[x0] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Mar 05 2013 *)
%o A106335 (PARI) A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2)); A106335=A106333/A106334
%Y A106335 Cf. A106333, A106334, A106336.
%K A106335 cons,nonn
%O A106335 0,1
%A A106335 _Paul D. Hanna_, Apr 29 2005