A226975 Decimal expansion I_1(1), the modified Bessel function of the first kind.

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, 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, 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

Offset

0,1

Comments

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

References

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

Links

Table of n, a(n) for n=0..107.

Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Formula

From Antonio Graciá 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

Cf. A002454, A197036.

Sequence in context: A020504 A293009 A011004 * A377635 A273065 A071629

Adjacent sequences: A226972 A226973 A226974 * A226976 A226977 A226978

Keyword

nonn,cons

Author

Horst-Holger Boltz, Jun 25 2013

Status

approved