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

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Apr 29 2005

Keywords

• cons
• nonn

Comments

The g.f. of A106336 equals (1/x)*Series_Reversion( x*eta(x)/eta(x^2)^2 ).

This constant is very close to 2^(3/2) / (3*sqrt(e*Pi)) = 0.3226276326921911330637735905807475397715626276499133673167401123748... - Vaclav Kotesovec, Aug 02 2017

Examples

			x/F(x)=0.322627632692191133096987138673983023322904243746717452160562...
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
so F(x) = 1.9873697211846841452692897833444126... (A106334)
at x = 0.6411803884299545796456448886283011... (A106333).
		

Crossrefs

Cf. A106333, A106334, A106336.

Programs

Mathematica

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 *)

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

Formula

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).