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%I #39 Feb 15 2024 18:28:02
%S 1,2,18,300,7350,238140,9604980,463783320,26087811750,1675417243500,
%T 120965124980700,9699203657543400,855146455806743100,
%U 82225620750648375000,8563211075317523625000,960221401912271649150000,115346595904711631854143750,14777934463556584363430887500
%N Coefficients in asymptotic expansion of I_0(x)sqrt(2*Pi*x)/e^x in powers of 1/(16x).
%C Central coefficients in exponential Riordan array [1/sqrt(1-2x), x]. - _Ralf Stephan_, Feb 07 2014
%D F. Bowman, Introduction to Bessel functions, Dover Publications Inc., New York 1958, see page 48. MR0097539
%D 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
%H Vincenzo Librandi, <a href="/A092563/b092563.txt">Table of n, a(n) for n = 0..200</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F E.g.f. A(x) = y satisfies: (8*x^2 - x)*y'' + (16*x - 1)*y + 2*y = 0.
%F G.f. A(x) = y satisfies: 8*x^2*y'' + (16*x-1)*y + 2*y = 0.
%F E.g.f.: F(1/2, 1/2;1;8x) = 1/AGM(1, (1-8x)^(1/2)).
%F a(n) = (2*n)!^2/(n!^3 * 2^n).
%F a(n)*2^n = A002894(n)*n!.
%F O.g.f.: hypergeom([1/2, 1/2], [], 8*x). - _Peter Luschny_, Oct 08 2015
%F E.g.f.: 2*K(8*x)/Pi, where K() is the complete elliptic integral of the first kind. - _Ilya Gutkovskiy_, Nov 23 2017
%F D-finite with recurrence: n*a(n) -2*(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Jan 23 2020
%e 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
%p H := hypergeom([1/2, 1/2], [], 8*x):
%p seq(coeff(series(H,x,20),x,n),n=0..16); # _Peter Luschny_, Oct 08 2015
%t Table[(2 n)!^2/n!^3/2^n, {n, 0, 30}] (* _Vincenzo Librandi_, Feb 08 2014 *)
%o (PARI) a(n)=if(n<0,0,(2*n)!^2/n!^3/2^n)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(1/agm(1,sqrt(1-8*x+x*O(x^n))),n))
%o (Magma) [Factorial(2*n)^2/Factorial(n)^3/2^n: n in [0..20]]; // _Vincenzo Librandi_, Feb 08 2014
%Y Cf. A002894.
%K nonn
%O 0,2
%A _Michael Somos_, Feb 28 2004
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