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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A115368 Decimal expansion of first zero of the Bessel function J_0(z).

Original entry on oeis.org

2, 4, 0, 4, 8, 2, 5, 5, 5, 7, 6, 9, 5, 7, 7, 2, 7, 6, 8, 6, 2, 1, 6, 3, 1, 8, 7, 9, 3, 2, 6, 4, 5, 4, 6, 4, 3, 1, 2, 4, 2, 4, 4, 9, 0, 9, 1, 4, 5, 9, 6, 7, 1, 3, 5, 7, 0, 6, 9, 9, 9, 0, 9, 0, 5, 9, 6, 7, 6, 5, 8, 3, 8, 6, 7, 7, 1, 9, 4, 0, 2, 9, 2, 0, 4, 4, 3, 6, 3, 4, 3, 7, 6, 0, 1, 4, 5, 2, 5, 4, 7, 8, 6, 8, 9
Offset: 1

Views

Author

Eric W. Weisstein, Jan 21 2006

Keywords

Comments

"This [constant] arises from the study of a vibrating, homogeneous membrane that is uniformly stretched across the unit disk. [Its square] is the principal frequency of the sound one hears when a kettledrum is struck." - Quoted from the book by Steven R. Finch.
Siegel proves (the Main Theorem) that J_0(z) is transcendental if z is algebraic and nonzero, but since in our case J_0(z) = 0 is not transcendental it follows that z cannot be algebraic. - Charles R Greathouse IV, Oct 20 2020

Examples

			2.4048255576957727686...

References

  • Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 7.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 221.
  • C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1. Translated as "On some applications
  • of Diophantine approximations" by Clemens Fuchs.

Links

Crossrefs

Cf. A115369, A115370, A115371, A115372, A115373.
Cf. A000275, A002190, A188489, A181167, A181168.

Programs

A181167 G.f.: 1 = Sum_{n>=0} a(n)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 1, 8, 165, 6384, 397320, 36273600, 4566166605, 757975618400, 160424015864112, 42164387189608320, 13473505313334666600, 5144136790654611953280, 2312696796696904699224000, 1209297981696245764641077760, 727688337054213932985609546525
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2010

Keywords

Comments

Compare g.f. to a g.f of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k>=0} C(2n+k,k)*(-x)^k.

Examples

			E.g.f.: E(x) = 1 + x^2/2! + 8*x^4/4! + 165*x^6/6! + 6384*x^8/8! +...
where the e.g.f. equals the continued fraction:
E(x) = 1/(1 - x^2/(2 - x^2/(3 - x^2/(4 - x^2/(5 - x^2/(6 - x^2/(7 - x^2/(8 - x^2/(9 - x^2/(10 - ...)))))))))). [Due to Matthieu Josuat-Vergès]
Illustrate the g.f. by the series:
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 8*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 165*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 6384*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 397320*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 36273600*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 4566166605*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
+...
Compare to a g.f. of the Catalan numbers (A000108):
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...)
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...)
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...)
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...)
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...)
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...)
+...
Surprisingly, terms a(n) are divisible by n*A000108(n) for n>0:
a(2)=2*2*2, a(3)=3*5*11, a(4)=4*14*114, a(5)=5*42*1892, a(6)=6*132*45800, a(7)=7*429*1520535, ..., a(n)=n*A000108(n)*A181168(n).

Links

Crossrefs

Cf. A181168, A180716.
Cf. A002190. [From Paul D. Hanna, Oct 09 2010]
Cf. A115368.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, x/(y+1)*(b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 08 2018
  • Mathematica
    nmax=20; Table[(CoefficientList[Series[BesselJ[1,2*x]/x/BesselJ[0,2*x],{x,0,2*nmax}],x] * Range[0,2*nmax]!)[[2*n-1]],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)
  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))}
  • PARI
    /* Formula: a(n) = A000108(n)*A002190(n+1) implies: */
{a(n)=binomial(2*n,n)/(n+1)*(n+1)!^2*polcoeff(-log(sum(m=0,n+1,(-x)^m/m!^2)+O(x^(n+2))),n+1)} \\ Paul D. Hanna, Oct 09 2010
  • PARI
    /* Continued Fraction expansion of the E.G.F.: */
{a(n)=local(CF=1+O(x));for(i=0,n,CF=1/((n-i+1)-x^2*CF));(2*n)!*polcoeff(CF,2*n)}

Formula

a(n) = n*A000108(n)*A181168(n) = C(2n,n-1)*A181168(n) for n>0, with a(0)=1.
a(n) = A000108(n)*A002190(n+1), where A002190 describes the coefficients in -log(BesselJ(0,2*sqrt(x))) and A000108 is the Catalan numbers. - Paul D. Hanna, Oct 09 2010
Differentiating -log(BesselJ(0,2*sqrt(x))) and substituting z=z^2 gives the e.g.f. Sum_{n>=0} a(n) * z^(2*n)/(2n)! = BesselJ(1,2*z)/z/BesselJ(0,2*z). Consequently, using Gauss' continued fraction, this e.g.f. is also: 1/(1-z^2/(2-z^2/(3-z^2/(4-z^2/(5-z^2/...))))). - Matthieu Josuat-Vergès, Apr 17 2011
E.g.f.: U(0) where U(k) = 1 - x^2/(x^2 - (k+1)*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 15 2012
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 16/BesselJZero[0,1]^2 = 2.76664110449031883070186935..., c = 4/(sqrt(Pi)*BesselJZero[0,1]^2) = 0.390227523142124366836071453... . - Vaclav Kotesovec, Jul 31 2014
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