A334383 -id:A334383 - cp's OEIS frontend

A070910 Decimal expansion of BesselI(0,2).

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

Offset: 1

Benoit Cloitre, May 20 2002

			2.2795853023360672674372044408115333532858411...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:5 at page 20.

Michael Penn, An exponential trigonometric integral., YouTube video, 2020.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A096789, A070913 (continued fraction), A006040.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), this sequence (I(0,2)).

RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* Robert G. Wilson v, Jul 09 2004 *)
(* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

besseli(0,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k>=0} 1/k!^2.
From Peter Bala, Aug 19 2013: (Start)
Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.
This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..oo} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)
From Amiram Eldar, May 29 2021: (Start)
Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.
Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)
Equals BesselJ(0,2*i). - Jianing Song, Sep 18 2021

A091681 Decimal expansion of BesselJ(0,2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 28 2004

The Pierce Expansion of this number is the squares > 1: 4,9,16,25,... - Franklin T. Adams-Watters, May 22 2006

			0.223890779...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Pierce Expansion

Cf A000290, A068985, A073701.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), this sequence (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[N[BesselJ[0, 2], 250]][[1]] (* G. C. Greubel, Dec 26 2016 *)

besselj(0,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k>=0} (-1)^k/(k!)^2.
Continued fraction expansion: BesselJ(0,2) = 1/(4 + 4/(8 + 9/(15 + ... + (n - 1)^2/(n^2 + 1 + ...)))). See A073701 for a proof. - Peter Bala, Feb 01 2015
Equals BesselI(0,2*i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 08 2011

			1.26606587775200833559824462521471753760767031135496...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Chapter 9.6.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A002454, A242282.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), this sequence (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

Maple
```
BesselI(0,1) ;evalf(%) ;
```

RealDigits[BesselJ[0, I], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

besseli(0,1) \\ Charles R Greathouse IV, Feb 19 2014

I_0(1) = Sum_{k>=0} 1/(4^k*k!^2) = Sum_{k>=0} 1/A002454(k).
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t)) dt.
Equals BesselJ(0,i). - Jianing Song, Sep 18 2021
From Amiram Eldar, Jul 09 2023: (Start)
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k)/(2^k*k!).
Equals e * Sum_{k>=0} (-1/2)^k * binomial(2*k,k)/k!. (End)

A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

This constant is transcendental.

			1/(4^0*0!^2) - 1/(4^1*1!^2) + 1/(4^2*2!^2) - 1/(4^3*3!^2) + ... = 0.765197686557966551449717526...

Cf. A000165, A002454.

Bessel function values: this sequence (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, 1], 10, 110] [[1]]

besselj(0, 1) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,1).

Equals BesselI(0,i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) + 1/(2^1*1!^2) + 1/(2^2*2!^2) + 1/(2^3*3!^2) + ... = 1.56608292975635053729238691...

Index entries for sequences related to Bessel functions or polynomials

Cf. A019774, A055546.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), this sequence (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselI[0, Sqrt[2]], 10, 110] [[1]]

suminf(k=0, 1/(2^k*(k!)^2)) \\ Michel Marcus, Apr 26 2020

besseli(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselI(0,sqrt(2)).

Equals BesselJ(0,sqrt(2)*i). - Jianing Song, Sep 18 2021

A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

Original entry on oeis.org

-1, 2, -16, 288, -9216, 460800, -33177600, 3251404800, -416179814400, 67421129932800, -13484225986560000, 3263182688747520000, -939796614359285760000, 317651255653438586880000, -124519292216147926056960000, 56033681497266566725632000000

Offset: 0

Eric W. Weisstein

sign

Coefficient of the Cayley-Menger determinant of order n.
A roller coaster has n rows of seats, each of which has room for two people.  |a(n)| is the number of ways n men and n women can be seated with a man and a woman in each row. - Geoffrey Critzer, Dec 17 2011
The o.g.f. of 1/a(n) is -BesselI(0,i*sqrt(2*x)), with i the imaginary unit. See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 10 2012
|a(n)|/2 is the number of integers k such that the digits of k and 2*k, written in base 2*n, are permutations of 0, 1, ..., 2*n-1. - Yifan Xie, Apr 12 2025

Andrew Howroyd, Table of n, a(n) for n = 0..100
Paul C. Kainen, Construction numbers: How to build a graph?, arXiv:2302.13186 [math.CO], 2023.
Usman A. Khan, Soummya Kar and Jose M. F. Moura, A novel geometric approach towards a linear theory for sensor localization, 2013.
Alan L. Mackay, On the regular heptagon, J. Math. Chemistry, vol. 21, 1997, 197-209.
Eric Weisstein's World of Mathematics, Cayley-Menger Determinant.

Cf. A000079, A001044, A019669, A334381, A334383.

Row of A340591 (in absolute values).

Mathematica
```
Table[(-1)^(n+1)2^n n!^2, {n, 0, 20}]
```

a(n)={(-1)^(n+1) * 2^n * n!^2} \\ Andrew Howroyd, Nov 07 2019

E.g.f.: -arcsinh(x/sqrt(2))^2. - Vladeta Jovovic, Aug 30 2004
Sum_{n>=0} |a(n)|/(2*n+1)! = Pi/2. - Daniel Suteu, Feb 06 2017
a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - Terry D. Grant, May 21 2017
From Amiram Eldar, Nov 18 2020: (Start)
Sum_{n>=0} 1/a(n) = (-1) * A334383.
Sum_{n>=0} (-1)^(n+1)/a(n) = A334381. (End)

Terms a(14) and beyond from Andrew Howroyd, Nov 07 2019

A348607 Decimal expansion of BesselJ(1,2).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.

Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

A367730 Decimal expansion of BesselJ(0,2/sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 28 2023

			0.69343678817918319009776046333354393...

Cf. A091681, A092615, A334380, A334383, A367729.

RealDigits[BesselJ[0, 2/Sqrt[3]], 10, 99][[1]]

besselj(0,2/sqrt(3)) \\ Michel Marcus, Nov 29 2023

Equals Sum_{k>=0} 1 / ((-3)^k * k!^2).

A385453 Decimal expansion of 6*Sum_{k>=0} (-1)^k/(k! (k + 3)! 2^k).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jun 29 2025

			0.881079450691092189890537005867857949397...

Michael I. Shamos, A catalog of the real numbers, (2007), p. 614.

Cf. A334383.

RealDigits[Hypergeometric0F1[4, -1/2], 10, 105][[1]]

Equals Hypergeometric0F1(4, -1/2).

Equals 12 * sqrt(2) * BesselJ(3, sqrt(2)).

cp's OEIS Frontend

A070910 Decimal expansion of BesselI(0,2).

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A091681 Decimal expansion of BesselJ(0,2).

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A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

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A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

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A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

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A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

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A348607 Decimal expansion of BesselJ(1,2).

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