A093601 Decimal expansion of constant in Kac's formula.

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

Offset

0,1

References

Kambiz Farahmand, Topics in Random Polynomials, Longman, 1998, pp. 34-35.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 141.

Links

Table of n, a(n) for n=0..101.

Mark Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc., Vol. 49, No. 4 (1943), pp. 314-320; correction, ibid., Vo. 49, No. 12 (1943), p. 938.

Eric Weisstein's World of Mathematics, Kac Formula.

Formula

From Amiram Eldar, Feb 13 2026: (Start)

Equals lim_{n->oo} z(n) - 2*log(n)/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n, with real coefficients independently chosen from a standard Gaussian distribution.

Equals (2/Pi) * (log(2) + Integral_{x>=0} (sqrt(1/x^2 - 1/sinh(x)^2) - 1/(x+1)) dx). (End)

Example

0.62573580720527001904647469180328602887812918031422...

Mathematica

(2/Pi)*(Log[2] + NIntegrate[Sqrt[1/x^2 - 4*(Exp[ -2*x]/(1 - Exp[ -2*x])^2)] - 1/(x + 1), {x, 0, Infinity}, WorkingPrecision -> 50, PrecisionGoal -> 20])

Crossrefs

Cf. A060294.

Sequence in context: A010371 A270614 A011489 * A297981 A193780 A220260

Adjacent sequences: A093598 A093599 A093600 * A093602 A093603 A093604

Keyword

nonn,cons

Author

Eric W. Weisstein, Apr 03 2004

Status

approved