A092563 Coefficients in asymptotic expansion of I_0(x)sqrt(2*Pi*x)/e^x in powers of 1/(16x).

1, 2, 18, 300, 7350, 238140, 9604980, 463783320, 26087811750, 1675417243500, 120965124980700, 9699203657543400, 855146455806743100, 82225620750648375000, 8563211075317523625000, 960221401912271649150000, 115346595904711631854143750, 14777934463556584363430887500

Offset

0,2

Comments

Central coefficients in exponential Riordan array [1/sqrt(1-2x), x]. - Ralf Stephan, Feb 07 2014

References

F. Bowman, Introduction to Bessel functions, Dover Publications Inc., New York 1958, see page 48. MR0097539

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 377. 9.7.1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

E.g.f. A(x) = y satisfies: (8*x^2 - x)*y'' + (16*x - 1)*y + 2*y = 0.

G.f. A(x) = y satisfies: 8*x^2*y'' + (16*x-1)*y + 2*y = 0.

E.g.f.: F(1/2, 1/2;1;8x) = 1/AGM(1, (1-8x)^(1/2)).

a(n) = (2*n)!^2/(n!^3 * 2^n).

a(n)*2^n = A002894(n)*n!.

O.g.f.: hypergeom([1/2, 1/2], [], 8*x). - Peter Luschny, Oct 08 2015

E.g.f.: 2*K(8*x)/Pi, where K() is the complete elliptic integral of the first kind. - Ilya Gutkovskiy, Nov 23 2017

D-finite with recurrence: n*a(n) -2*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jan 23 2020

Example

I_0(x)sqrt(2*Pi*x)/e^x ~ 1+2/(16x)+18/(16x)^2+300/(16x)^3+... where I_0(x) is a Bessel function

Maple

H := hypergeom([1/2, 1/2], [], 8*x):
seq(coeff(series(H, x, 20), x, n), n=0..16); # Peter Luschny, Oct 08 2015

Mathematica

Table[(2 n)!^2/n!^3/2^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 08 2014 *)

Prog

(PARI) a(n)=if(n<0, 0, (2*n)!^2/n!^3/2^n)
(PARI) a(n)=if(n<0, 0, n!*polcoeff(1/agm(1, sqrt(1-8*x+x*O(x^n))), n))
(Magma) [Factorial(2*n)^2/Factorial(n)^3/2^n: n in [0..20]]; // Vincenzo Librandi, Feb 08 2014

Crossrefs

Cf. A002894.

Sequence in context: A123385 A121564 A224384 * A258922 A192555 A370941

Adjacent sequences: A092560 A092561 A092562 * A092564 A092565 A092566

Keyword

nonn

Author

Michael Somos, Feb 28 2004

Status

approved