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

1,1

Links

Table of n, a(n) for n=1..105.

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.

Formula

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

Example

2.279585302336...

Mathematica

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]]

Prog

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

Crossrefs

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)).

Sequence in context: A155063 A324666 A011022 * A189040 A267214 A107386

Adjacent sequences: A070907 A070908 A070909 * A070911 A070912 A070913

Keyword

cons,easy,nonn

Author

Benoit Cloitre, May 20 2002

Status

approved