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.

Previous Showing 31-35 of 35 results.

A069660 Order of the subgroup of the symmetric group S_n generated by the cycles (1,3) and (1,2,3,...,n).

Original entry on oeis.org

6, 8, 120, 72, 5040, 1152, 362880, 28800, 39916800, 1036800, 6227020800, 50803200, 1307674368000, 3251404800, 355687428096000, 263363788800, 121645100408832000, 26336378880000, 51090942171709440000, 3186701844480000, 25852016738884976640000, 458885065605120000
Offset: 3

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 16 2002

Keywords

Crossrefs

Cf. A000984, A070734, A000142, A009445, A001044.

Programs

  • Mathematica
    a[n_] := If[OddQ[n], n!, 2 * ((n/2)!)^2]; Array[a, 20, 3] (* Amiram Eldar, Jul 12 2025 *)
  • PARI
    a(n) = if(n % 2, n!, 2 * ((n/2)!)^2); \\ Amiram Eldar, Jul 12 2025

Formula

If n is odd a(n) = n!, if n is even a(n) = 2 * ((n/2)!)^2 = 2 * n! / A000984(n/2) = 2 * A001044(n/2).
Sum_{n>=3} 1/a(n) = BesselI(0, 2)/2 + sinh(1) - 2. - Amiram Eldar, Jul 12 2025

Extensions

More terms from Benoit Cloitre, May 20 2002

A122685 a(n) = n! except that a(2) = -2 and a(2n) = 0 for n > 2.

Original entry on oeis.org

1, 1, -2, 6, 0, 120, 0, 5040, 0, 362880, 0, 39916800, 0, 6227020800, 0, 1307674368000, 0, 355687428096000, 0, 121645100408832000, 0, 51090942171709440000, 0, 25852016738884976640000, 0, 15511210043330985984000000, 0, 10888869450418352160768000000, 0, 8841761993739701954543616000000
Offset: 0

Views

Author

Paul Curtz, Jul 28 2007

Keywords

Comments

Same as A005212 for n > 2. - Georg Fischer, Oct 21 2018

Crossrefs

Cf. A005212, A009445.

Extensions

Edited by N. J. A. Sloane, Jul 29 2007

A127488 a(n) = (n^2)!/(2*(n!)).

Original entry on oeis.org

6, 30240, 435891456000, 64630041847212441600000, 258328699159653623241666283438080000000
Offset: 2

Views

Author

Artur Jasinski, Jan 16 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374, A134374

Programs

  • Mathematica
    Table[(n^2)!/(2(n!)), {n, 2, 6}]

Formula

a(n) ~ n^(2*n^2 - n + 1/2) / (2 * exp(n*(n-1))). - Vaclav Kotesovec, Oct 26 2017

A322379 Triangle T(s,d) read by rows: the number of 2-connected labeled cubic graphs with s simple edges and d double edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 120, 0, 0, 0, 0, 5040, 0, 0, 180, 0, 0, 362880, 1, 0, 0, 23520, 0, 0, 39916800, 0, 180, 0, 0, 3628800, 0, 0, 6227020800, 0, 0, 45360, 0, 0, 718502400, 0, 0, 1307674368000, 70, 0, 0, 13003200, 0, 0, 181621440000, 0, 0, 355687428096000, 0, 45360, 0, 0, 4340952000, 0, 0, 57537672192000, 0, 0
Offset: 0

Views

Author

R. J. Mathar, Dec 05 2018

Keywords

Examples

			The triangle starts
  0;
  0, 0;
  0, 0,  6;
  0, 0,  0,   120;
  0, 0,  0,     0, 5040;
  0, 0,180,     0,    0, 362880;
  1, 0,  0, 23520,    0,      0, 39916800;

Links

Crossrefs

Cf. A009445 (diagonal), A007099 (left column).

Programs

  • Maple
    # expand g(s,d) of eq (21) of Chae et al.
g2x := 6*x^5/4! ;
for itr from 1 to 16 do
    g2xx := expand(diff(g2x,x)) ;
    g2x := (x^5-x^8)*g2x*g2xx+(x^4-2*x^7+x^10+x^5*y-x^8*y)/2*g2xx
        +(2*x^4+x^7)*g2x^2
        +(8*x^3-6*x^6-x^9+x^12+2*x*y-2*x^4*y+8*x^7*y-2*x^10*y)/2*g2x
        + x^5/4 -3*x^8/4 +3*x^11/4-x^14/4 +3*x^6*y/2-9*x^9*y/4+3*x^12*y/4+x*y^2/2
        -x^4*y^2+7*x^7*y^2/4-x^10*y^2/2 ;
    g2x := expand(%) ;
    g2x := taylor(g2x,x=0,itr+5) ;
    g2x := convert(g2x,polynom) ;
    g2 := expand(int(g2x,x)) ;
    for s from 0 to itr+1 do
        g := coeftayl(g2,x=0,s) ;
        for d from 0 to s do
            twon := (2*s+4*d)/3 ;
            coeftayl(g,y=0,d) ;
            printf("%a,",%*twon!) ;
        end do:
        printf("\n") ;
    end do:
end do:

Formula

T(3s,0) = A007099(s).

A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

Original entry on oeis.org

1, 1, 2, 4, 6, 24, 36, 48, 120, 720, 576, 720, 1440, 5040, 40320, 14400, 17280, 30240, 80640, 362880, 3628800, 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600, 25401600, 29030400, 43545600, 87091200, 239500800, 958003200, 6227020800, 87178291200
Offset: 0

Views

Author

Peter Luschny, Feb 27 2024

Keywords

Examples

			Triangle starts:
[0]      1;
[1]      1,      2;
[2]      4,      6,     24;
[3]     36,     48,    120,     720;
[4]    576,    720,   1440,    5040,   40320;
[5]  14400,  17280,  30240,   80640,  362880,  3628800;
[6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;

Crossrefs

Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).
Cf. A119502, A143084.

Programs

  • Maple
    T := (n, k) -> (n - k)! * (n + k)!:
seq(seq(T(n, k), k = 0..n), n = 0..7);
  • Mathematica
    Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 05 2024 *)

Formula

Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.
Previous Showing 31-35 of 35 results.

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

Content is available under The OEIS End-User License Agreement.