A298595 G.f.: Sum_{n>=0} a(n)*x^(2*n)/((2*n)!)^2 = 1/BesselJ(0,x).

1, 1, 27, 4275, 2326275, 3260434275, 9824561849025, 56272951734424425, 560476093710119461875, 9074718916938795106861875, 226586114542199918676706160625, 8362768986063791790897266120885625, 440616849129306857329147873116900455625, 32189976281042425371050387695609814928515625

Offset: 0

Author

Ilya Gutkovskiy, Jan 22 2018

Keywords

• nonn

Examples

			1/BesselJ(0,x) = 1 + x^2/(2!)^2 + 27*x^4/(4!)^2 + 4275*x^6/(6!)^2 + 2326275*x^8/(8!)^2 + ...
		

Links

• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A000275, A001818, A002454.

Programs

Mathematica

nmax = 13; Table[(CoefficientList[Series[1/BesselJ[0, x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 13; Table[(CoefficientList[Series[1/Hypergeometric0F1[1, -x^2/4], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = ((2*n)!)^2 * [x^(2*n)] 1/BesselJ(0,x).

a(n) ~ c * Pi * 2^(4*n+3) * n^(4*n+1) / (exp(4*n) * r^(2*n+1)), where r = BesselJZero(0, 1) = A115368 = 2.40482555769... and c = 1 / BesselJ(1, r) = 1.9262348469772531439976485375138638... - Vaclav Kotesovec, May 04 2024