A334380 -id:A334380 - 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)

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

Original entry on oeis.org

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000

Offset: 0

N. J. A. Sloane

Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.

T. D. Noe, Table of n, a(n) for n = 0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.

Cf. A000165, A001818, A079484, A197036, A334380.

J1: A002474, J2: A002506, J3: A014401.

[4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019

Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)

a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019

(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022

A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 0

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) + ... = 0.5591341444189799174882684679...

Cf. A055546, A092605.

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

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

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

Equals BesselJ(0,sqrt(2)).

Equals BesselI(0,sqrt(2)*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

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)!).

A387195 Decimal expansion of J_0(Pi/3), Bessel Function of the first kind of index 0 at A019670.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 21 2025

Because Pi/3 is near 1, this constant is near A334380.

			J_0(Pi/3) = 0.744071970752929535741...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A019670, A387193, A387194.

First[RealDigits[BesselJ[0, Pi/3], 10, 100]] (* Paolo Xausa, Aug 21 2025 *)

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).

A344074 Decimal expansion of Bessel Y_0(1).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, May 08 2021

			0.08825696421567695798292676...

Index entries for sequences related to Bessel functions or polynomials

Cf. A334380, A046961.

Maple
```
Digits:=100; evalf(BesselY(0,1));
```

Join[{0},First[RealDigits[N[BesselY[0,1],86]]]] (* Stefano Spezia, May 09 2021 *)

bessely(0, 1) \\ Michel Marcus, May 09 2021

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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A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

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A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

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

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A348607 Decimal expansion of BesselJ(1,2).

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