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.

A052856 E.g.f.: (1-3*exp(x)+exp(2*x))/(exp(x)-2).

Original entry on oeis.org

1, 2, 4, 14, 76, 542, 4684, 47294, 545836, 7087262, 102247564, 1622632574, 28091567596, 526858348382, 10641342970444, 230283190977854, 5315654681981356, 130370767029135902, 3385534663256845324
Offset: 0

Views

Author

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

Keywords

Comments

Previous name was: A simple grammar.
Stirling transform of A005212(n-1)=[1,1,0,6,0,120,0,...] is a(n-1)=[1,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n*A052612(n-1)=[0,2,-2,12,-24,...] is a(n-1)=[0,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n)=[2,2,6,24,120,...] is a(n)=[2,2,4,14,76,...]. - Michael Somos, Mar 04 2004

Links

Crossrefs

A000670(n)=a(n)-1, if n>0. A032109(n)=a(n)/2, if n>0.
A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

Programs

  • Maple
    spec := [S,{B=Sequence(C),C=Set(Z,1 <= card),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nn=20},CoefficientList[Series[(1-3Exp[x]+Exp[x]^2)/(-2+Exp[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 24 2012 *)
  • PARI
    a(n)=if(n<0,0,n!*polcoeff(subst(y+1/(1-y),y,exp(x+x*O(x^n))-1),n))

Formula

E.g.f.: (1-3*exp(x)+exp(x)^2)/(-2+exp(x))
a(n) ~ n!/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 05 2013

Extensions

New name using e.g.f., Vaclav Kotesovec, Oct 05 2013

A191662 a(n) = n! / A000034(n-1).

Original entry on oeis.org

1, 1, 6, 12, 120, 360, 5040, 20160, 362880, 1814400, 39916800, 239500800, 6227020800, 43589145600, 1307674368000, 10461394944000, 355687428096000, 3201186852864000, 121645100408832000, 1216451004088320000, 51090942171709440000, 562000363888803840000
Offset: 1

Views

Author

Paul Curtz, Jun 10 2011

Keywords

Comments

The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
A005408 = (2*n+1)/1 = 1, 3, 5, 7, 9, ... (k=1)
A000290 = (n^2)/1 = 1, 4, 9, 16, 25, ... (k=2)
A000330 = n*(n+1)*(2*n+1)/6 = 1, 5, 14, 30, 55, ... (k=3)
A002415 = (n^2)*(n^2-1)/12 = 1, 6, 20, 50, 105, ... (k=4)
A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27, 77, 182, ... (k=5)
A040977 = (n^2)*(n^2-1)*(n^2-4)/360 = 1, 8, 35, 112, 294, ... (k=6)
A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
The first superdiagonal of this array appears in A029651. - Paul Curtz, Jul 04 2011
The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - Johannes W. Meijer, Jun 22 2011

Crossrefs

Cf. A000142, A009445, A002674, A064680.
Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
Cf. A052612.

Programs

Formula

a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
a(n+1) = A064680(n+1) * a(n).
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - Georg Fischer, Nov 25 2022
a(n) = A052612(n)/2 for n >= 1. - Alois P. Heinz, Sep 05 2023

Extensions

More terms from Harvey P. Dale, Mar 14 2014

A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

Original entry on oeis.org

2, 3, 12, 30, 240, 840, 10080, 45360, 725760, 3991680, 79833600, 518918400, 12454041600, 93405312000, 2615348736000, 22230464256000, 711374856192000, 6758061133824000, 243290200817664000, 2554547108585472000, 102181884343418880000, 1175091669949317120000
Offset: 1

Views

Author

N. J. A. Sloane, Mar 14 2015

Keywords

Comments

Table 2 in chapter 7 of the preprint contains a typo: a(9) is not 725860. - R. J. Mathar, Mar 14 2015

Links

Crossrefs

Cf. A052612, A052616, A052849, A098558, A208529, A256032, A174549 (bisection).

Programs

  • Maple
    A256031 := proc(n)
    if type(n,'odd') then
        2*n! ;
    else
        (n+1)*(n-1)! ;
    end if;
end proc:
seq(A256031(n),n=1..20) ; # R. J. Mathar, Mar 14 2015
  • Mathematica
    a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];
Array[a, 20] (* Jean-François Alcover, Nov 24 2017, from Maple *)

Formula

There are simple formulas for the two bisections - see Dolinka et al.
a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - Omar E. Pol, Mar 14 2015
Sum_{n>=1} 1/a(n) = (e^2+3)/(4*e) = 1/e + sinh(1)/2. - Amiram Eldar, Feb 02 2023

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000
Offset: 1

Views

Author

M. F. Hasler, Apr 22 2015

Keywords

Comments

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Links

Crossrefs

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

Programs

  • Magma
    [Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015
  • Maple
    A256881 := n!/round(n/2);
  • Mathematica
    Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)
  • PARI
    A256881(n)=n!/round(n/2)

Formula

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Extensions

Definition clarified by Robert Israel, Sep 03 2018
Showing 1-4 of 4 results.

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

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