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

0,1

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

Links

Table of n, a(n) for n=0..104.

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).

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A106333, A106334, A106336.

Sequence in context: A007567 A093055 A285733 * A218698 A352836 A065474

Adjacent sequences: A106332 A106333 A106334 * A106336 A106337 A106338

Keyword

cons,nonn

Author

Paul D. Hanna, Apr 29 2005

Status

approved