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

Author

Michael Somos, Feb 28 2004

Keywords

• nonn

Comments

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

Examples

			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
		

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].

Crossrefs

Cf. A002894.

Programs

Magma

[Factorial(2*n)^2/Factorial(n)^3/2^n: n in [0..20]]; // Vincenzo Librandi, Feb 08 2014

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 *)

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))

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