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 1-2 of 2 results.

A126115 E.g.f.: sqrt(1+2*x)/(1-2*x).

Original entry on oeis.org

1, 3, 11, 69, 537, 5475, 64755, 916965, 14536305, 263680515, 5239150875, 115916048325, 2768235849225, 72290366223075, 2016224400665475, 60700190066641125, 1936215798778886625, 66023235942444655875, 2370503834057244760875, 90300788789652000685125, 3603830757053442135845625
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Comments

Old name: Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x).
Denominators are successive powers of 2.

Examples

			The fractions are 1, 3/2, 11/4, 69/8, 537/16, 5475/32, 64755/64, 916965/128, ...

Links

Crossrefs

Cf. A001147, A126119.

Programs

  • Maple
    f:= gfun:-rectoproc({-2*(n+1)*(1+2*n)*a(n)-3*a(n+1)+a(n+2), a(0)=1,a(1)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Mar 12 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[Sqrt[1+2x]/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Sep 21 2018 *)

Formula

b(n) = a(n)/n! satisfies b(n) = (3*b(n-1) + 2*(2*n-3)*b(n-2))/n, b(0)=1, b(1)=3. - Sergei N. Gladkovskii, Jul 22 2012, corrected by Robert Israel, Mar 12 2018
D-finite with recurrence: a(n+2) = (2*(n+1))*(1+2*n)*a(n)+3*a(n+1). - Robert Israel, Mar 12 2018
E.g.f.: sqrt(1+2*x)/(1-2*x). - Sergei N. Gladkovskii, Jul 22 2012

Extensions

Better name by Sergei N. Gladkovskii, Jul 22 2012
Edited by Robert Israel, Mar 12 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
Showing 1-2 of 2 results.

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

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