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A070910 Decimal expansion of BesselI(0,2). 38
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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..infinity} 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
KEYWORD
cons,easy,nonn
AUTHOR
Benoit Cloitre, May 20 2002
STATUS
approved

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Last modified May 22 23:49 EDT 2024. Contains 372758 sequences. (Running on oeis4.)