cp's OEIS Frontend

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

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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

A181168 G.f.: 1 = 1/(1+x) + Sum_{n>=1} a(n)*C(2n,n-1)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 2, 11, 114, 1892, 45800, 1520535, 66256610, 3666164264, 251038266192, 20835983387100, 2060833345614120, 239466622145739120, 32297762247056413536, 5003953730422122499023, 882564184814509784837250
Offset: 1

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

			Illustrate the g.f. by the series:
1 = (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*1*1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 +...)
+ 2*2*2*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 +...)
+ 3*5*11*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 +...)
+ 4*14*114*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 5*42*1892*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 6*132*45800*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 7*429*1520535*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4+..) +...
which indicates a connection of this sequence to the Catalan numbers.

Links

Crossrefs

Cf. A181167, A000108, A115368.

Programs

  • 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]] / Binomial[2n,n-1],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)
  • PARI
    {a(n)=if(n<1, 0, ((-1)^(n-1)-polcoeff(sum(m=0, n-1, a(m)*binomial(2*m, m-1)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))/binomial(2*n, n-1))}

Formula

a(n) = A181167(n)/C(2n,n-1) for n>=1.
a(n) ~ (n!)^2 * (2/BesselJZero[0,1])^(2*n+2), where BesselJZero[0,1] = A115368 = 2.40482555769... . - Vaclav Kotesovec, Jul 31 2014

A295944 Expansion of e.g.f. 1/(1 - x/(1 - x^2/(2 - x^3/(3 - x^4/(4 - x^5/(5 - x^6/(6 - x^7/(7 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 9, 48, 330, 2760, 26670, 295680, 3686760, 51080400, 778516200, 12944131200, 233156523600, 4522777459200, 94000269963600, 2083918752115200, 49086474041404800, 1224240044169542400, 32229413145084355200, 893129953569780326400, 25987602379142314310400, 792175050968260985625600
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2017

Keywords

Links

Crossrefs

Cf. A005169, A181167, A257544, A295945.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k, k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ c * d^n * n!, where d = 1.38558212161941692858602713469062337279193542118277136584639901149123656221... and c = 0.53969028910223464320214486945875671476165137860949073877514057198146... - Vaclav Kotesovec, Sep 24 2020
Showing 1-3 of 3 results.

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