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 1-4 of 4 results.

A213067 E.g.f.: arctan(cos(x)^2) - Pi/4.

Original entry on oeis.org

0, -1, -2, 44, 1408, -18016, -5095232, -139605376, 56961507328, 8306292414464, -1178066937638912, -640316054325354496, -7088737339266301952, 76268423227563817631744, 18895160315230467816030208, -12297988177132848140606242816
Offset: 0

Views

Author

Olivier Gérard, Jun 04 2012

Keywords

Comments

This function is even, with constant term Pi/4 = 0.785398163397...
It was missing from OEIS entries by Patrick Demichel.

Examples

			arctan(cos(x)^2) - Pi/4 = 0 - x^2/2 - 2*x^4/4! + 44*x^6/6! + 1408*x^8/8! + ...

Links

Crossrefs

Cf. A213066, A213068, A213069.

Programs

  • Mathematica
    Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@
  CoefficientList[Series[ArcTan[Cos[x]^2] - Pi/4, {x, 0, 30}], x] // ExpandAll
With[{nn=30},Take[CoefficientList[Series[ArcTan[Cos[x]^2]-Pi/4,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Feb 24 2022 *)

Formula

E.g.f.: arctan(cos(x)^2) - Pi/4.

A213069 Expansion of e.g.f. arcsinh(cos(x)*sech(x)), even powers only.

Original entry on oeis.org

0, -1, 3, -1, -77, -13921, 791043, 23892959, -3518362637, -801110007361, 149920222346883, 24069808471917119, -7334638751184472397, -2673575321959933341601, 1059696929013386749787523, 413637485668406346391368479
Offset: 0

Views

Author

Olivier Gérard, Jun 04 2012

Keywords

Comments

This function is even, with constant term arcsinh(1) = 0.881373587019543...

Examples

			(arcsinh(cos(x)*sech(x))-arcsinh(1))/sqrt(2) = -x^2/2 + 3*x^4/4! - x^6/6! - 77*x^8/8! + ...

Links

Crossrefs

Cf. A213066, A213067, A213068.

Programs

  • Mathematica
    Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@ CoefficientList[Series[(ArcSinh[Cos[x]*Sech[x]] - ArcSinh[1])/Sqrt[2], {x, 0, 30}], x] // ExpandAll
With[{nn=30},Take[CoefficientList[Series[(ArcSinh[Cos[x]Sech[x]]-ArcSinh[ 1])/ Sqrt[2],{x,0,nn}],x]Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Mar 24 2013 *)

Formula

E.g.f.: (arcsinh(cos(x)*sech(x))-arcsinh(1))/sqrt(2).

A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

Original entry on oeis.org

2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000, 15944266600786427904000, 637770664031457116160000
Offset: 0

Views

Author

Emeric Deutsch

Keywords

Comments

Partition the set {1,2,...,n+2} into an even number of subsets. Arrange (linearly order) the elements within each subset and then arrange the subsets. - Geoffrey Critzer, Mar 03 2010

Links

Crossrefs

Essentially the same as A052564.
Cf. A088312.

Programs

  • GAP
    List([0..20], n-> 2^n*Factorial(n+2)); # G. C. Greubel, May 05 2019
  • Magma
    [2^n*Factorial(n+2): n in [0..20]]; // G. C. Greubel, May 05 2019
  • Maple
    seq(count(Permutation(n+1))*count(Composition(n)),n=1..17); # Zerinvary Lajos, Oct 16 2006
  • Mathematica
    Drop[CoefficientList[Series[(1-x)^2/(1-2x), {x, 0, 20}], x]* Table[n!, {n, 0, 20}], 2] (* Geoffrey Critzer, Mar 03 2010 *)
Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[2/(1 - 2*x)^3, {x, 0, 20}], x]// ExpandAll (* Vincenzo Librandi, Jan 04 2013 - after Olivier Gérard in A213068 *)
Table[n!Binomial[n+2,2]2^(n+1),{n,0,30}] (* Harvey P. Dale, Dec 27 2022 *)
  • PARI
    a(n) = (n+2)!*2^n; \\ Joerg Arndt, May 05 2019
  • Sage
    [2^n*factorial(n+2) for n in (0..20)] # G. C. Greubel, May 05 2019

Formula

a(n) = Sum_{k=0..n} (n+2)!*C(n,k).
Prepend the sequence with 1,0, then e.g.f. is (1-x)^2/(1-2*x). - Geoffrey Critzer, Mar 03 2010
E.g.f.: 2/(1-2*x)^3. - R. J. Mathar, Nov 27 2011
a(n) = 2*A051578(n). - R. J. Mathar, Apr 26 2017
a(n) = (n+2)! * 2^n. - Joerg Arndt, May 05 2019
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = 4*sqrt(e) - 6.
Sum_{n>=0} (-1)^n/a(n) = 4/sqrt(e) - 2. (End)

A052564 Expansion of e.g.f. x*(1-x)/(1-2*x).

Original entry on oeis.org

0, 1, 2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Partition the set {1,2,...,n} into an odd number of subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010

Links

Crossrefs

Essentially the same as A014297.

Programs

  • Magma
    [n le 1 select n else 2^(n-2)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 05 2019
  • Maple
    spec := [S,{S=Prod(Z,Sequence(Prod(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    a = x/(1 - x); CoefficientList[Series[a/(1 - a^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Geoffrey Critzer, Mar 05 2010 *)
Part[#,Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[x*(1-x)/(1-2x), {x, 0, 20}], x]// ExpandAll (* Vincenzo Librandi, Jan 04 2013 - after Olivier Gérard in A213068 *)
With[{nn=20},CoefficientList[Series[x (1-x)/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2025 *)
  • PARI
    {a(n) = if(n<=1, n, 2^(n-2)*n!)}; \\ G. C. Greubel, May 05 2019
  • PARI
    my(x='x+O('x^20)); concat([0], Vec(serlaplace(x*(1-x)/(1-2*x)))) \\ Felix Fröhlich, May 05 2019
  • Sage
    [0,1]+[2^(n-2)*factorial(n) for n in (2..20)] # G. C. Greubel, May 05 2019

Formula

E.g.f.: x*(1-x)/(1-2*x).
a(n) = 2*n*a(n-1), with a(0)=0, a(1)=1, a(2)=2.
a(n) = 2^(n-2) * n! for n>1.
a(n) = A002866(n) - A014297(n-2) for n>1. - Geoffrey Critzer, Mar 05 2010
Showing 1-4 of 4 results.

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