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 1-3 of 3 results.

A145219 a(n) is the number of even permutations (of an n-set) with exactly 1 fixed point.

Original entry on oeis.org

1, 0, 0, 8, 15, 144, 910, 7440, 66717, 667520, 7342236, 88107480, 1145396395, 16035550608, 240533257770, 3848532125984, 65425046139705, 1177650830517120, 22375365779822392, 447507315596451240, 9397653627525472071, 206748379805560390160, 4755212735527888968390
Offset: 1

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Author

Abdullahi Umar, Oct 09 2008

Keywords

Examples

			a(4) = 8 because there are exactly 8 even permutations (of a 4-set) having 1 fixed point, namely: (123), (132), (124), (142), (134), (143), (234), (243).

Links

Crossrefs

Cf. A003221, A145220, A145222.

Programs

  • PARI
    x = 'x + O('x^30); Vec(serlaplace((x*(1-x^2/2) * exp(-x))/(1-x))) \\ Michel Marcus, Apr 04 2016

Formula

a(n) = n*A003221(n-1), (n > 0).
E.g.f.: (x*(1-x^2/2) * exp(-x))/(1-x).
D-finite with recurrence (-3*n+7)*a(n) +(n-2)*(3*n-10)*a(n-1) +(9*n-8)*(n-3)*a(n-2) +2*(3*n-2)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 06 2023

Extensions

More terms from Alois P. Heinz, Nov 19 2013

A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

Original entry on oeis.org

0, 0, 6, 0, 90, 420, 3780, 33264, 333900, 3670920, 44054010, 572697840, 8017775766, 120266628300, 1924266063720, 32712523068960, 588825415259640, 11187682889909904, 223753657798227150, 4698826813762734240, 103374189902780197170, 2377606367763944481780
Offset: 2

Views

Author

Abdullahi Umar, Oct 09 2008

Keywords

Examples

			a(4) = 6 because there are exactly 6 odd permutations (of a 4-set) having 2 fixed points, namely: (12), (13), (14), (23), (24), (34).

Links

Crossrefs

Cf. A000387 (odd permutations with no fixed points), A145222 (odd permutations with exactly 1 fixed point), A145220 (even permutations with exactly 2 fixed points).

Programs

  • Maple
    egf:= x^4 * exp(-x)/(4*(1-x));
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 01 2011
  • Mathematica
    A000387[n_] := Subfactorial[n-2] Binomial[n, 2];
a[n_] := (n(n-1)/2) A000387[n-2];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 30 2025 *)
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ Michel Marcus, Apr 04 2016

Formula

a(n) = A145225(n,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).
E.g.f.: x^4*exp(-x)/(4*(1-x)).
D-finite with recurrence +(-n+6)*a(n) +(n-2)*(n-7)*a(n-1) +(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

More terms from Alois P. Heinz, Feb 01 2011

A145224 Triangle read by rows: T(n,k) is the number of even permutations (of an n-set) with exactly k fixed points.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 8, 0, 0, 1, 24, 15, 20, 0, 0, 1, 130, 144, 45, 40, 0, 0, 1, 930, 910, 504, 105, 70, 0, 0, 1, 7413, 7440, 3640, 1344, 210, 112, 0, 0, 1, 66752, 66717, 33480, 10920, 3024, 378, 168, 0, 0, 1
Offset: 0

Views

Author

Abdullahi Umar, Oct 09 2008

Keywords

Examples

			Triangle starts:
   1;
   0,  1;
   0,  0,  1;
   2,  0,  0, 1;
   3,  8,  0, 0, 1;
  24, 15, 20, 0, 0, 1;
  ...

Links

Crossrefs

Row sums give A001710.
Columns k=0..2 are A003221, A145219, A145220.
Cf. A008290, A055137.

Formula

T(n,k) = C(n,k)*A003221(n-k).
E.g.f.: (x^k(1-x^2/2) e^(-x))/k!(1-x).
T(n,k) + A145225(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023
T(n,k) = (A008290(n,k) + A055137(n,k))/2. - Julian Hatfield Iacoponi, Aug 08 2024
Showing 1-3 of 3 results.

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