OFFSET
0,1
COMMENTS
This is also the derivative of the zeroth modified Bessel function at 1.
LINKS
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind
FORMULA
From Antonio GraciĆ” Llorente, Jan 29 2024: (Start)
I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).
Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k+1)/(2^k*k!).
Equals (-e) * Sum_{k>=0} (-1/2)^k * binomial(2*k,k+1)/k!
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t))*cos(t) dt. (End)
EXAMPLE
0.56515910399248502720769602760986330732889962162109...
MATHEMATICA
RealDigits[BesselI[1, 1], 10, 110][[1]]
PROG
(PARI) besseli(1, 1) \\ Charles R Greathouse IV, Feb 19 2014
(SageMath)
((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)
# Peter Luschny, Jan 29 2024
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Horst-Holger Boltz, Jun 25 2013
STATUS
approved