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.

A052560 a(n) = 3*n!.

Original entry on oeis.org

3, 3, 6, 18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400, 1437004800, 18681062400, 261534873600, 3923023104000, 62768369664000, 1067062284288000, 19207121117184000, 364935301226496000, 7298706024529920000, 153272826515128320000
Offset: 0

Views

Author

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

Keywords

Comments

a(n) is the size of the centralizer of a 3-cycle in the symmetric group S_(n+3). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
3 times factorial numbers. - Omar E. Pol, Jan 17 2009

Links

Crossrefs

Cf. A000142, A129380.
Row 13 of A276955 (from term a(3)=18 onward).

Programs

Formula

E.g.f.: 3/(1-x).
a(n) = n*a(n-1), with a(0) = 3.
For n>0: a(n) = Sum_{k=1..n+1} A083746(k). - Reinhard Zumkeller, Apr 14 2007

A082425 a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).

Original entry on oeis.org

1, 1, 5, 27, 169, 1217, 9939, 90871, 920069, 10222989, 123698167, 1619321459, 22805443881, 343835923129, 5525934478859, 94309281772527, 1703461402016269, 32465970250192421, 651123070017747999, 13707854105636799979, 302258183029291439537, 6966331456484621749329
Offset: 1

Views

Author

Benoit Cloitre, Apr 24 2003

Keywords

Links

Crossrefs

Cf. A001339, A056543, A083746.
Cf. A007808, A074143, A082427, A082428, A082430, A383436, A383437.

Programs

  • Magma
    [n le 2 select 1 else (n^2*Self(n-1) +1)/(n-1): n in [1..30]]; // G. C. Greubel, Feb 03 2024
  • Maple
    a:= n -> n*n!*add(1/(k*(k-1)*k!), k = 2..n): seq(a(n), n = 2..20); # Peter Bala, Jul 09 2008
  • Mathematica
    a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j,n-1}]];
Table[a[n], {n,40}] (* G. C. Greubel, Feb 03 2024 *)
  • SageMath
    @CachedFunction # a = A082425
def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1,n))
[a(n) for n in range(1,41)] # G. C. Greubel, Feb 03 2024

Formula

For n >= 2, a(n) = floor(n*(3-e)*n!).
a(n) = n*A056543(n) - 1, n > 1. - Vladeta Jovovic, Apr 26 2003
From Peter Bala, Jul 09 2008: (Start)
In the following remarks we use an offset of 1, i.e., a(1) = 1, a(2) = 1, a(3) = 5, ... .
For n >= 2, a(n) = n*n!*Sum_{k = 2..n} 1/(k*(k-1)*k!).
For n >= 2, a(n) = 3*n*n! - Sum_{k = 0..n} (k+1)!*binomial(n,k).
Limit_{n -> oo} a(n)/(n*n!) = 3 - e.
E.g.f.: 1 + t + (3*t - exp(t))/(1-t)^2.
a(n) = A083746(n+2) - A001339(n).
Recurrence relation: a(1) = 1, a(2) = 1, a(3) = 5, a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n >= 4.
Recurrence relation: a(1) = 1, a(2) = 1, a(n) = (n^2*a(n-1) + 1)/(n-1) for n >= 3.
The recurrence relation x(n) = (n^2*x(n-1) - 1)/(n-1), for n >= 2, has the general solution x(n) = n*n!*x(1) - a(n); particular solutions are A007808 (x(1) = 1) and A001339 (x(1) = 3). (End)

Extensions

Offset corrected by G. C. Greubel, Feb 03 2024

A052673 a(n) = 3*n*n!.

Original entry on oeis.org

0, 3, 12, 54, 288, 1800, 12960, 105840, 967680, 9797760, 108864000, 1317254400, 17244057600, 242853811200, 3661488230400, 58845346560000, 1004293914624000, 18140058832896000, 345728180109312000
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A083746, A122972.
Cf. A000142, A008585.

Programs

  • Magma
    [3*(Factorial(n+1)-Factorial(n)): n in [0..30]]; // G. C. Greubel, Jun 12 2022
  • Maple
    spec := [S,{S=Prod(Sequence(Z),Sequence(Z),Union(Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Table[3 n n!,{n,0,20}] (* Harvey P. Dale, Feb 12 2017 *)
  • SageMath
    [3*n*factorial(n) for n in (0..30)] # G. C. Greubel, Jun 12 2022

Formula

E.g.f.: 3*x/(1-x)^2.
Recurrence: a(0)=0, a(1)=3, (n-1)*a(n) = n^2*a(n-1).
a(n) = A122972(n+2) - A122972(n) for n > 0. - Reinhard Zumkeller, Sep 21 2006
For n>0: a(n) = A083746(n+2). - Reinhard Zumkeller, Apr 14 2007
G.f.: 3*Hypergeometric2F0([2,2], [], x). - G. C. Greubel, Jun 12 2022

A129379 a(n) = sum of sums of all sets of three distinct preceding terms otherwise a(n) = n for n <= 3.

Original entry on oeis.org

1, 2, 3, 6, 36, 288, 3360, 55440, 1241856, 36427776, 1358235648, 62818398720, 3531789972480, 237336286150656, 18792718657929216, 1732062236305809408, 183865068161693614080, 22273939685873740677120
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 14 2007

Keywords

Links

Crossrefs

Cf. A000217, A083746, A129380.

Programs

  • Magma
    A129379:= func< n | n le 3 select Binomial(n,Floor(n/2)) else (3/2^(n-3))*Binomial(n-2,2)*(&*[k^2-k+2: k in [0..n-3]])  >;
[A129379(n): n in [1..30]]; // G. C. Greubel, Feb 02 2024
  • Mathematica
    a[n_]:= a[n]= If[n<5, Binomial[n, Floor[n/2]], (n-2)*(n^2-7*n+14)*a[n- 1]/(2*(n-4))];
Table[a[n], {n,40}] (* G. C. Greubel, Feb 02 2024 *)
Round[Flatten[{{1, 2, 3}, Table[3 * 2^(3-n) * (n-3) * (n-2) * Cosh[Sqrt[7]*Pi/2] * Gamma[n - 5/2 - I*Sqrt[7]/2] * Gamma[n - 5/2 + I*Sqrt[7]/2]/Pi, {n, 4, 20}]}]] (* Vaclav Kotesovec, Feb 03 2024 *)
  • SageMath
    def A129379(n): return binomial(n,n//2) if n<4 else 3*binomial(n-2,2)*product(j^2-j+2 for j in range(n-2))//2^(n-3)
[A129379(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2024

Formula

a(n) = (1/2)*(n-2)*(n-3)*Sum_{j=1..n-1} a(j) for n > 3, with a(1) = 1, a(2) = 2, a(3) = 3.
a(n) = A000217(n-3)*A129380(n-1) for n > 3.
From G. C. Greubel, Feb 02 2024: (Start)
a(n) = (6/2^(n-4))*binomial(n-2,2)*|Pochhammer((3+i*sqrt(7))/2, n-4)|^2 for n > 3, otherwise a(n) = n.
a(n) = (3/2^(n-3))*binomial(n-2,2)*Product_{k=0..n-3} (k^2 - k + 2), for n > 3, otherwise a(n) = n.
a(n) = (n-2)*(n^2-7*n+14)/(2*(n-4))*a(n-1), for n > 4, otherwise a(n) = binomial(n, floor(n/2)).
(End)
From Vaclav Kotesovec, Feb 03 2024: (Start)
For n>=4, a(n) = 3 * 2^(3-n) * (n-3) * (n-2) * cosh(sqrt(7)*Pi/2) * Gamma(n - 5/2 - i*sqrt(7)/2) * Gamma(n - 5/2 + i*sqrt(7)/2)/Pi, where i is the imaginary unit.
a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)
Showing 1-4 of 4 results.

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