%I #6 Feb 12 2013 10:24:27

%S 6,4,1,1,8,0,3,8,8,4,2,9,9,5,4,5,7,9,6,4,5,6,4,4,8,8,8,6,2,8,3,0,1,1,

%T 0,6,5,5,3,4,1,9,6,1,8,9,1,0,0,7,1,1,9,0,8,7,7,5,6,0,3,0,5,0,5,1,3,1,

%U 7,2,7,8,4,5,7,5,9,2,4,7,3,3,2,3,7,8,4,6,3,5,1,2,0,8,8,3,7,9,3,2,2,4,8,9,6

%N Decimal expansion of the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

%C Not equal to exp(-4/9), which agrees with the first 16 decimal places. Related to Jacobi theta constant theta_2 and Dedekind's eta(x^2)^2/eta(x): Sum_{n>=0} x^(n*(n+1)/2) = 1.9873697... (A106334). This constant divided by constant A106334 equals constant A106335, the radius of convergence of the g.f. of A106336.

%F Sum_{n>=0} (1 - n*(n+1)/2)*x^(n*(n+1)/2) = 0.

%e 0 = 1 - 2*x^3 - 5*x^6 - 9*x^10 - 14*x^15 - 20*x^21 - 27*x^28 - ...

%e x=0.641180388429954579645644888628301106553419618910071190877560305051317278

%t digits = 105; g[x_?NumericQ] := NSum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 100]; x /. FindRoot[g[x], {x, 1/2}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 12 2013 *)

%o (PARI) solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2)))

%Y Cf. A106334, A106335, A106332, A106336.

%K cons,nonn

%O 0,1

%A _Paul D. Hanna_, Apr 29 2005