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.

Showing 1-10 of 10 results.

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

Original entry on oeis.org

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)

References

  • Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
  • Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
  • E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.

Links

Crossrefs

Cf. A000165, A001818, A079484, A197036, A334380.
J1: A002474, J2: A002506, J3: A014401.

Programs

Formula

(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022

A002474 Denominators of coefficients of odd powers of x of the expansion of Bessel function J_1(x).

Original entry on oeis.org

2, 16, 384, 18432, 1474560, 176947200, 29727129600, 6658877030400, 1917756584755200, 690392370511872000, 303772643025223680000, 160391955517318103040000, 100084580242806496296960000, 72861574416763129304186880000, 61203722510081028615516979200000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The corresponding numerators are A033999(n) = (-1)^n.

Examples

			a(3) = 18432 = 128*6*24, since J_{1}(x) = x/2 - x^3/16 + x^5/384 - x^7/18432 + ...

References

  • Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:2 and 52:6:3 at page 483, 513.

Links

Crossrefs

Cf. J_0: A002454, J_2: A002506, J_3: A014401, J_4: A061403, J_5: A061404, J_6: A061405, J_7: A061407, J_9: A061440 J_10: A061441.
Cf. A010790, A033999.

Programs

  • Magma
    [2^(2*n+1)*Factorial(n)*Factorial(n+1): n in [0..30]]; // G. C. Greubel, Sep 21 2024
  • Maple
    a:= n-> denom(coeff(series(BesselJ(1, x), x, 2*n+2), x, 2*n+1)):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 21 2024
  • Mathematica
    CoefficientList[Series[BesselJ[1,x], {x,0,30}], x][[2 ;; ;; 2]]//Denominator
Table[2^(2*n+1)*n!*(n+1)!, {n,0,30}] (* G. C. Greubel, Sep 21 2024 *)
  • PARI
    a(n) = n!^2 * (n+1) << (2*n+1) \\ Charles R Greathouse IV, Oct 23 2023
  • PARI
    first(n)=my(x='x+O('x^(2*n+1)),t=besselj(1,x)); vector(n+1,k,2*denominator(polcoeff(t,2*k-2))) \\ Charles R Greathouse IV, Oct 23 2023
  • SageMath
    [2^(2*n+1)*factorial(n)*factorial(n+1) for n in range(31)] # G. C. Greubel, Sep 21 2024

Formula

a(n) = 2^(2n+k) * n! * (n+k)! here for k=1, i.e., Bessel's J1(x) has the denominator a(n) for the coefficient of x^(2*n+1), n >= 0.
a(n) = 2^(2n+1)*A010790(n).

Extensions

Name specified, numerators given, formula augmented by Wolfdieter Lang, Aug 25 2015

A002506 Denominators of coefficients of expansion of Bessel function J_2(x).

Original entry on oeis.org

8, 96, 3072, 184320, 17694720, 2477260800, 475634073600, 119859786547200, 38355131695104000, 15188632151261184000, 7290543432605368320000, 4170190843450270679040000, 2802368246798581896314880000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(2) = 3072 = 64*2*24, J2(x) = x^2/8 - x^4/96 + x^6/3072 - x^8/184320 +- ...

References

  • Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7

Links

Crossrefs

J0: A002454, J1: A002474, J3: A014401.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[2,x],{x,0,30}],x],{3,-1,2}]] (* Harvey P. Dale, Sep 21 2013 *)

Formula

a(n) = 2^(2n+k) * n! * (n+k)! here for k=2, i.e., Bessel's J2(x).
a(n) - 4*n*(n+2)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013

Extensions

Previous Mathematica program corrected by Harvey P. Dale, Sep 21 2013

A061403 Denominators in the series for Bessel function J4(x).

Original entry on oeis.org

384, 7680, 368640, 30965760, 3963617280, 713451110400, 171228266496000, 52738306080768000, 20251509535014912000, 9477706462386978816000, 5307515618936708136960000, 3502960308498227370393600000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[4, x], {x, 0, 500}], x], {5, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061404 Denominators in the series for Bessel function J5(x).

Original entry on oeis.org

3840, 92160, 5160960, 495452160, 71345111040, 14269022208000, 3767021862912000, 1265719345938432000, 526539247910387712000, 265375780946835406848000, 159225468568101244108800000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[5, x], {x, 0, 500}], x], {6, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061405 Denominators in the series for Bessel function J6(x).

Original entry on oeis.org

46080, 1290240, 82575360, 8918138880, 1426902220800, 313918488576000, 90408524709888000, 32908702994399232000, 14743098941490855936000, 7961273428405062205440000, 5095214994179239811481600000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[6, x], {x, 0, 500}], x], {7, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061407 Denominators in the series for Bessel function J8(x).

Original entry on oeis.org

10321920, 371589120, 29727129600, 3923981107200, 753404372582400, 195885136871424000, 65817405988798464000, 27643310515295354880000, 14153374983831221698560000, 8661865490104707679518720000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[8, x], {x, 0, 50}], x], {9, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061440 Denominators in the series for Bessel function J9(x).

Original entry on oeis.org

185794560, 7431782400, 653996851200, 94175546572800, 19588513687142400, 5484783832399872000, 1974522179663953920000, 884585936489451356160000, 481214749450261537751040000, 311827157643769476462673920000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[9, x], {x, 0, 50}], x], {10, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061441 Denominators in the series for Bessel function J10(x).

Original entry on oeis.org

3715891200, 163499212800, 15695924428800, 2448564210892800, 548478383239987200, 164543514971996160000, 63184709749246525440000, 30075921840641346109440000, 17323730980209415359037440000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[10, x], {x, 0, 50}], x], {11, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061406 Denominators in the series for Bessel function J7(x).

Original entry on oeis.org

645120, 20643840, 1486356480, 178362777600, 31391848857600, 7534043725824000, 2350621642457088000, 921443683843178496000, 442292968244725678080000, 254760749708961990574080000
Offset: 0

Views

Author

Frank Ellermann, Jun 11 2001

Keywords

Links

Crossrefs

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[BesselJ[7, x], {x, 0, 50}], x], {8, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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