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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A143165 Expansion of the exponential generating function arcsin(2*x)/(2*(1-2*x)^(3/2)).

Original entry on oeis.org

0, 1, 6, 49, 468, 5469, 73362, 1138005, 19737000, 383284665, 8163588510, 190709475705, 4818820261500, 131650382056725, 3850053335966250, 120466494638624925, 4002649276431128400, 141156781966460192625, 5252646220794868029750, 206149276075766825426625
Offset: 0

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Author

Wolfdieter Lang, Sep 15 2008

Keywords

Comments

Used in A024199(n+1) = A003148(n) + a(n).
Binomial convolution of [0,1^2,0,2^2,0,...,0,((2*k)!/k!)^2,0,...] (e.g.f. arcsin(2*x)/2) with the double factorials A001147.

Examples

			a(3) + A003148(3) = 49 + 27 = 76 = A024199(4).

Links

Crossrefs

Cf. A003148, A024199.

Programs

  • Maple
    f:= gfun:-rectoproc({2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0, a(0)=0,a(1)=1,a(2)=6},a(n),remember):
map(f, [$0 .. 30]); # Robert Israel, Feb 07 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[ArcSin[2x]/(2(1-2x)^(3/2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 18 2019 *)
  • PARI
    x = 'x + O('x^40); concat(0, Vec(serlaplace(asin(2*x)/(2*(1-2*x)^(3/2))))) \\ Michel Marcus, Jun 18 2017

Formula

E.g.f.: arcsin(2*x)/(2*(1-2*x)^(3/2)).
a(n) = sum(binomial(n,2*k+1)*(4^k)*((2*k-1)!!)^2*(2*(n-2*k)-1)!!,k=0..floor(n/2)), with (2*n-1)!!:= A001147(n) (double factorials).
a(n) ~ Pi * 2^(n-1/2) * n^(n+1) / exp(n) * (1 - sqrt(2/(Pi*n))). - Vaclav Kotesovec, Mar 18 2014
2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0. - Robert Israel, Feb 07 2018

A126121 Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x)^2.

Original entry on oeis.org

1, 5, 31, 255, 2577, 31245, 439695, 7072695, 127699425, 2562270165, 56484554175, 1358576240175, 35374065613425, 992016072172125, 29792674421484975, 954480422711190375, 32479589325536978625, 1170329273010701929125, 44502357662442514209375, 1781390379962467540641375
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Comments

Denominators are successive powers of 2.

Examples

			The fractions are 1, 5/2, 31/4, 255/8, 2577/16, 31245/32, 439695/64, ...

Links

Crossrefs

Cf. A003148, A001174, A126115, A126119.

Programs

  • Mathematica
    With[{nn=20},Numerator[CoefficientList[Series[Sqrt[1+x]/(1-x)^2,{x, 0, nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Jan 29 2016 *)
  • PARI
    x='x+O('x^25); Vec(serlaplace(sqrt(1+2*x)/(1-2*x)^2)) \\ G. C. Greubel, May 25 2017

Formula

E.g.f.: 1/G(0) where G(k) = 1 - 4*x/(1 + x/(1 - x - (2*k+1)/( 2*k+1 - 4*(k+1)*x/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 28 2012
From Benedict W. J. Irwin, May 19 2016: (Start)
E.g.f.: sqrt(1+2*x)/(1-2*x)^2.
a(n) = (-1)^(n+1)*2^(n-1)*(n-3/2)!*2F1(2,-n;(3/2)-n;-1)/sqrt(Pi).
(End)
D-finite with recurrence a(n) -5*a(n-1) -2*(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 08 2021
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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