A106333 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).

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Apr 29 2005

Keywords

• cons
• nonn

Comments

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.

Examples

			0 = 1 - 2*x^3 - 5*x^6 - 9*x^10 - 14*x^15 - 20*x^21 - 27*x^28 - ...
x=0.641180388429954579645644888628301106553419618910071190877560305051317278
		

Crossrefs

Cf. A106334, A106335, A106332, A106336.

Programs

Mathematica

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 *)

PARI

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

Formula

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