A244258 -id:A244258 - cp's OEIS frontend

A271948 Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.

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

Offset: 0

Jean-François Alcover, Apr 20 2016

			0.049469852279228075333485464056253836603725107670028013295315781039...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Xavier Gourdon, Largest component in random combinatorial structures, Discrete Mathematics 180, 1998, Pages 185-209.

Cf. A084945, A143297, A244067, A244258, A244261, A261873, A271871.

digits = 96; F[x_] := 1 - Exp[-x]/Sqrt[Pi*x] - Erf[Sqrt[x]]; Clear[f, g];
f[m_] := f[m] = 2 NIntegrate[(1 - (1 - F[x])^-1), {x, 0, m}, WorkingPrecision -> digits + 10]; f[m = 100]; f[m = 2 m]; Print["m = ", m]; While[RealDigits[f[m], 10, digits + 5][[1]] != RealDigits[f[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
g[m_] := g[m] = (8/3) NIntegrate[(1 - (1 - F[x])^-1)*x, {x, 0, m}, WorkingPrecision -> digits + 10]; g[m = 100]; g[m = 2 m]; Print["m = ", m]; While[RealDigits[g[m], 10, digits + 5][[1]] != RealDigits[g[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
Join[{0}, RealDigits[g[m] - f[m]^2, 10, digits][[1]]]

A244261 Decimal expansion of c = 2.4149..., a random mapping statistics constant such that the asymptotic expectation of the maximum rho length (graph diameter) in a random n-mapping is c*sqrt(n).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 24 2014

			2.4149010237176162411...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.
P. Flajolet and A. M. Odlyzko, Random Mapping Statistics, Advances in Cryptology - EUROCRYPT '89, J.-J. Quisquater and J. Vandewalle (eds.), Lecture Notes in Computer Science, Springer Verlag, 1990, pp. 329-354.

Cf. A084945, A244067, A244258.

evalf(sqrt(Pi/2)*Int(1 - exp(Ei(-x) - Int((exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))), y=0..x)), x=0..infinity)); # Vaclav Kotesovec, Aug 12 2019

digits = 20; m0 = 100; dm = 10; I0[x_?NumericQ] := NIntegrate[(Exp[-y]/y)*(1 - Exp[-2*(y/(Exp[x - y] - 1))]), {y, 0, x}, WorkingPrecision -> digits+5]; Clear[f]; f[m_] := f[m] = Sqrt[Pi/2]* NIntegrate[1 - Exp[ExpIntegralEi[-x] - I0[x]], {x, 0, m}, WorkingPrecision -> digits+5]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], Print["m = ", m]; m = m + dm]; RealDigits[f[m], 10, digits] // First

I(x) = integral_(0..x) (exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))) dy,

c = sqrt(Pi/2)*integral_(0..infinity) 1 - exp(Ei(-x) - I(x)) dx, where Ei is the exponential integral function.

A271871 Decimal expansion of a constant related to the expected number of vertices of the largest tree associated with a random mapping on n symbols.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 20 2016

			0.48349834715442550092402636085075619444924667954133810432926494155247...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Xavier Gourdon, Largest component in random combinatorial structures, Discrete Mathematics 180, 1998, Pages 185-209.

Cf. A084945, A143297, A244067, A244258, A244261, A261873.

digits = 98; F[x_] := 1 - Exp[-x]/Sqrt[Pi*x] - Erf[Sqrt[x]]; Clear[f]; f[m_] := f[m] = 2 NIntegrate[1-(1-F[x])^-1, {x, 0, m}, WorkingPrecision -> digits+10]; f[m = 100]; f[m = 2 m]; Print["m = ", m]; While[ RealDigits[ f[m], 10, digits + 5][[1]] != RealDigits[f[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]]; RealDigits[f[m/2], 10, digits + 5][[1]]

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A271948 Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.

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A244261 Decimal expansion of c = 2.4149..., a random mapping statistics constant such that the asymptotic expectation of the maximum rho length (graph diameter) in a random n-mapping is c*sqrt(n).

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A271871 Decimal expansion of a constant related to the expected number of vertices of the largest tree associated with a random mapping on n symbols.

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