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 A092563 Coefficients in asymptotic expansion of I_0(x)sqrt(2*Pi*x)/e^x in powers of 1/(16x). 2
 1, 2, 18, 300, 7350, 238140, 9604980, 463783320, 26087811750, 1675417243500, 120965124980700, 9699203657543400, 855146455806743100, 82225620750648375000, 8563211075317523625000, 960221401912271649150000 (list; graph; refs; listen; history; text; internal format)
 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 A350366 Adjacent sequences: A092560 A092561 A092562 * A092564 A092565 A092566 KEYWORD nonn AUTHOR Michael Somos, Feb 28 2004 STATUS approved

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