A087197 -id:A087197 - cp's OEIS frontend

A243961 Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.

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

Offset: 1

Jean-François Alcover, Jun 16 2014

According to Steven Finch, no exact expression of this moment mu(8) is known, unlike the moments mu(n) for n<8.

			1.423600306045277753078324649306257253...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197 mu(2), A243446 mu(3), A243448 mu(4), A243453 mu(5), A243523 mu(6), A243524 mu(7).

digits = 100; m0 = 5; dm = 5; f[x_] := 1/ Sqrt[2*Pi]*Exp[-x^2/2]; F[x_] := 1/2*Erf[x/Sqrt[2]] + 1/2; Clear[mu8]; mu8[m_] := mu8[m] = 8*NIntegrate[x*F[x]^7*f[x], {x , -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; mu8[m0]; mu8[m = m0 + dm]; While[RealDigits[mu8[m]] != RealDigits[mu8[m - dm]], Print["m = ", m]; m = m + dm]; RealDigits[mu8[m], 10, digits] // First

integral_(-infinity..infinity) 8*x*F(x)^7*f(x) dx, where f(x) is the normal (0,1) density and F(x) its cumulative distribution.

A351401 Decimal expansion of erfi(1)/e, where erfi is the imaginary error function.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 10 2022

The alternating sum of reciprocals of the factorials of the positive half-integers.

			0.60715770584139372911503823580074492116122092866515...

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler Functions, Related Topics and Applications, New York, NY: Springer, 2020. See p. 94, eq. (4.12.9.6).
Constantin Milici, Gheorghe Drăgănescu, and J. Tenreiro Machado, Fractional Differential Equations, Introduction to Fractional Differential Equations, Springer, Cham, 2019. See p. 12, eq. (1.9).

Eric Weisstein's World of Mathematics, Erfi.
Eric Weisstein's World of Mathematics, Mittag-Leffler Function.
Wikipedia, Mittag-Leffler function.

Cf. A001113, A001147, A068985, A087197, A099288, A122803, A351400.

evalf(exp(-1)*erfi(1), 120);  # Alois P. Heinz, Feb 10 2022

Mathematica
```
RealDigits[Erfi[1]/E, 10, 100][[1]]
```

real(-I*(1.0-erfc(I)))/exp(1) \\ Michel Marcus, Feb 10 2022

Equals Sum_{k>=0} (-1)^k/(k + 1/2)! = Sum_{k>=1} (-1)^(k+1)/Gamma(k + 1/2).
Equals E_{1, 3/2}(-1), where E_{a,b}(z) is the two-parameter Mittag-Leffler function.
Equals (-1/sqrt(Pi)) * Sum_{k>=1} (-2)^k/(2*k-1)!!.
Equals A068985 * A099288.

A119575 a(n) = binomial(2n,n)(n+3)^2/(n+1).

Original entry on oeis.org

9, 16, 50, 180, 686, 2688, 10692, 42900, 173030, 700128, 2838524, 11522056, 46802700, 190182400, 772913160, 3141129780, 12764118870, 51857916000, 210638666700, 855355383960, 3472419702180, 14092569803520, 57176602275000, 231908298827400, 940340123399196, 3811765978738368

Offset: 0

Zerinvary Lajos, May 31 2006

Cf. A000984, A087197.

[seq (binomial(2*n,n)*(n+3)^2/(n+1),n=0..25)];

a[n_] := Binomial[2*n, n]*(n + 3)^2/(n + 1); Table[a[n], {n, 0, 25}] (* Robert P. P. McKone, Aug 25 2023 *)

a(n) = binomial(2*n,n)/(n+1)*(n+3)^2 \\ Charles R Greathouse IV, Oct 23 2023

From Stefano Spezia, Aug 24 2023: (Start)
O.g.f.: (2*(sqrt(1 - 4*x) - 1) + x*(21 - 8*sqrt(1 - 4*x) - 50*x))/(x*(1 - 4*x)^(3/2)).
E.g.f.: exp(2*x)*((9 + 2*x)*BesselI(0, 2*x) + 2*(x - 2)*BesselI(1, 2*x)).
a(n) ~ c*4^n*sqrt(n), where c = A087197. (End)

More terms from Stefano Spezia, Aug 24 2023

A289503 Decimal expansion of 3/sqrt(Pi).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 07 2017

			1.6925687506432688608442383546823177...

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E10.

Steven R. Finch, Mean width of a regular simplex, arxiv:1111.4976 [math.MG], (2016), mu_3.
Index entries for transcendental numbers.

Cf. A087197 (1/sqrt(Pi)).

Maple
```
3/sqrt(Pi); evalf(%) ;
```

First[RealDigits[3/Sqrt[Pi], 10, 100]] (* Paolo Xausa, Apr 29 2024 *)

3/sqrt(Pi) \\ Charles R Greathouse IV, Oct 01 2022

from mpmath import mp, sqrt, pi
mp.dps=90
print([int(z) for z in list(str(3/sqrt(pi)).replace('.', '')[:-2])]) # Indranil Ghosh, Jul 08 2017

Equals 3*A087197.

A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.54969627774735302956219538317088212891969758...

Decimal expansions of Integral_{x=0..oo} 1 / sqrt(1 + x^k) dx: A118292 (k=3), A093341 (k=4), this sequence (k=5).

Cf. A001622, A087197, A340723, A352324.

RealDigits[Gamma[3/10] Gamma[6/5]/Sqrt[Pi], 10, 105][[1]]
RealDigits[2^(2/5) * Gamma[1/5]^2 / (5*GoldenRatio*Gamma[2/5]), 10, 105][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

Equals Gamma(3/10) * Gamma(6/5) / sqrt(Pi).

Equals 2^(2/5) * Gamma(1/5)^2 / (5 * phi * Gamma(2/5)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 09 2024

A378942 Decimal expansion of (1/sqrt(Pi) + e*erfc(-1))/2.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 11 2024

			2.7865848321550198766289520248877912002691926567823...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.6, p. 262.

Cf. A001113, A019739, A087197, A087198, A099286, A222392.

RealDigits[(1/Sqrt[Pi]+E Erfc[-1])/2,10,100][[1]]

Equals (1 + e*sqrt(Pi)*(1 + erf(1)))/(2*sqrt(Pi)).

Equals A222392 / 2. - Amiram Eldar, Feb 15 2025

cp's OEIS Frontend

A243961 Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.

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A351401 Decimal expansion of erfi(1)/e, where erfi is the imaginary error function.

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A119575 a(n) = binomial(2*n,n)*(n+3)^2/(n+1).

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A289503 Decimal expansion of 3/sqrt(Pi).

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A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

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A378942 Decimal expansion of (1/sqrt(Pi) + e*erfc(-1))/2.

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A119575 a(n) = binomial(2n,n)(n+3)^2/(n+1).