A226975 - OEIS

%I #45 Feb 16 2025 08:33:20

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

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

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

%N Decimal expansion I_1(1), the modified Bessel function of the first kind.

%C This is also the derivative of the zeroth modified Bessel function at 1.

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function of the First Kind</a>.

%F From _Antonio Graciá Llorente_, Jan 29 2024: (Start)

%F I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).

%F Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.

%F Equals exp(-1) * Sum_{k>=0} binomial(2*k,k+1)/(2^k*k!).

%F Equals (-e) * Sum_{k>=0} (-1/2)^k * binomial(2*k,k+1)/k!

%F Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t))*cos(t) dt. (End)

%e 0.56515910399248502720769602760986330732889962162109...

%t RealDigits[BesselI[1, 1], 10, 110][[1]]

%o (PARI) besseli(1,1) \\ _Charles R Greathouse IV_, Feb 19 2014

%o (SageMath)

%o ((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)

%o # _Peter Luschny_, Jan 29 2024

%Y Cf. A002454, A197036.

%K nonn,cons,changed

%O 0,1

%A _Horst-Holger Boltz_, Jun 25 2013