A247398 Decimal expansion of a constant 'v' such that the asymptotic variance of the distribution of the longest cycle given a random n-permutation evaluates as v*n^2.
0, 3, 6, 9, 0, 7, 8, 3, 0, 0, 6, 4, 8, 5, 2, 2, 0, 2, 1, 7, 7, 1, 0, 7, 0, 0, 2, 9, 2, 9, 3, 2, 7, 6, 4, 0, 2, 2, 4, 6, 2, 2, 3, 3, 1, 0, 5, 8, 6, 8, 5, 1, 9, 6, 4, 7, 6, 2, 2, 7, 8, 2, 0, 7, 3, 0, 4, 8, 9, 1, 9, 4, 7, 1, 5, 3, 0, 8, 0, 6, 2, 8, 5, 1, 1, 8, 9, 3, 0, 4, 4, 9, 1, 0, 3, 4, 3
Offset: 0
Examples
0.03690783006485220217710700292932764...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.
Links
- Eric Weisstein's MathWorld, Golomb-Dickman Constant
- Wikipedia, Golomb-Dickman constant
Crossrefs
Cf. A084945.
Programs
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Maple
evalf(int((x-exp(Ei(-x))*x),x=0..infinity) - int( (1-exp(Ei(-x))),x=0..infinity)^2, 50); # Vaclav Kotesovec, Aug 12 2019
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Mathematica
v = NIntegrate[x - E^ExpIntegralEi[-x]*x, {x, 0, Infinity}, WorkingPrecision -> 80] - NIntegrate[1 - E^ExpIntegralEi[-x], {x, 0, Infinity}, WorkingPrecision -> 80]^2; Join[{0}, RealDigits[v, 10, 40] // First]
Formula
v = integral_{0..infinity} x-e^Ei(-x)*x dx - (integral_{0..infinity} 1-e^Ei(-x) dx)^2, where Ei is the exponential integral function. [corrected by Vaclav Kotesovec, Aug 12 2019]
Extensions
More digits from Vaclav Kotesovec, Aug 12 2019