OFFSET
0,2
COMMENTS
Central coefficients in exponential Riordan array [1/sqrt(1-2x), x]. - Ralf Stephan, Feb 07 2014
REFERENCES
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
F. Bowman, Introduction to Bessel functions, Dover Publications Inc., New York 1958, see page 48. MR0097539
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
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 28 2004
STATUS
approved