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 1-10 of 12 results. Next

A134375 a(n) = (n!)^4.

Original entry on oeis.org

1, 1, 16, 1296, 331776, 207360000, 268738560000, 645241282560000, 2642908293365760000, 17340121312772751360000, 173401213127727513600000000, 2538767161403058526617600000000, 52643875858853821607942553600000000, 1503561738404723998944447273369600000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Comments

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma_4(gcd(i,j)) for 1 <= i,j <= n, and n>0, where sigma_4 is A001159. - Enrique Pérez Herrero, Aug 13 2011

Links

Crossrefs

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374.
Row n=4 of A225816.

Programs

  • Maple
    a:= n-> (n!)^4:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013
  • Mathematica
    Table[((n)!)^(4), {n, 0, 10}]

Formula

a(n) = det(S(i+4,j), 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013

A134374 a(n) = ((2n+1)!)^2.

Original entry on oeis.org

1, 36, 14400, 25401600, 131681894400, 1593350922240000, 38775788043632640000, 1710012252724199424000000, 126513546505547170185216000000, 14797530453474819213543604224000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134375, A334378.

Programs

Formula

a(n) = A009445(n)^2 = A001044(2n+1). - Wesley Ivan Hurt, May 02 2014
From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334378.
Sum_{n>=0} (-1)^n/a(n) = Im(BesselJ(0, 2*exp(3*Pi*i/4))). (End)

A134368 a(n) = ((2n)!)^(n+1).

Original entry on oeis.org

1, 4, 13824, 268738560000, 106562062388507443200000, 2283380023591730815784976384000000000000, 5785737804304645733190746102656048717392091545600000000000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A134367, A134368, A134369, A134370, A134371, A134374, A134375.

Programs

  • Mathematica
    Table[((2n)!)^(n + 1), {n, 0, 10}]

Formula

a(n) ~ 2^((n+1)*(2*n+1)) * exp(1/24 - 2*n*(n+1)) * n^((n+1)*(4*n+1)/2) * Pi^((n+1)/2). - Vaclav Kotesovec, Oct 26 2017

A134366 a(n) = (n!)^(n-1).

Original entry on oeis.org

1, 1, 2, 36, 13824, 207360000, 193491763200000, 16390160963076096000000, 173238200573946282828103680000000, 300679807141675805997423113304381849600000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A134366. A134367, A134368, A134369, A134370, A134371, A134374, A134375.

Programs

  • Maple
    a:=n->mul(n!/k, k=1..n): seq(a(n), n=0..9); # Zerinvary Lajos, Jan 22 2008
restart:with (combinat):a:=n->mul(stirling1(n,1), j=3..n): seq(a(n), n=1..10); # Zerinvary Lajos, Jan 01 2009
  • Mathematica
    Table[(n!)^(n - 1), {n, 0, 10}]
  • PARI
    a(n) = (n!)^(n-1); \\ Michel Marcus, Dec 23 2015

Formula

a(n) ~ exp(1/12 + n - n^2) * n^((n-1)*(2*n+1)/2) * (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Oct 26 2017

Extensions

Offset corrected to 0 by Michel Marcus, Dec 23 2015

A134369 a(n) = ((2n+1)!)^(n+1).

Original entry on oeis.org

1, 36, 1728000, 645241282560000, 6292383221978976013516800000, 4045146997974190235742848547815424000000000000, 363046466970952735968096996065196818096105852014637875200000000000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134370, A134371, A134374, A134375.

Programs

  • Mathematica
    Table[((2n+1)!)^(n + 1), {n, 0, 10}]

Formula

a(n) ~ 2^(2*(n+1)^2) * exp(13/24 - 2*n*(n+1)) * n^((n+1)*(4*n+3)/2) * Pi^((n+1)/2). - Vaclav Kotesovec, Oct 26 2017

A134371 a(n) = ((2n)!)^n.

Original entry on oeis.org

1, 2, 576, 373248000, 2642908293365760000, 629238322197897601351680000000000, 12078744213598964456884373878200091017216000000000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134370, A134374, A134375.

Programs

  • Mathematica
    Table[((2n)!)^(n), {n, 0, 10}]

Formula

a(n) ~ 2^(n*(2*n+1)) * exp(1/24 - 2*n^2) * n^(n*(4*n+1)/2) * Pi^(n/2). - Vaclav Kotesovec, Oct 26 2017

A134372 a(n) = ((2n)!)^2.

Original entry on oeis.org

1, 4, 576, 518400, 1625702400, 13168189440000, 229442532802560000, 7600054456551997440000, 437763136697395052544000000, 40990389067797283140009984000000, 5919012181389927685417441689600000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134374, A134375, A334379, A334632.

Programs

  • Mathematica
    Table[((2n)!)^(2), {n, 0, 10}]
((2*Range[0,20])!)^2 (* Harvey P. Dale, Jul 14 2011 *)
  • PARI
    a(n) = ((2*n)!)^2; \\ Michel Marcus, Nov 16 2020

Formula

From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334379.
Sum_{n>=0} (-1)^n/a(n) = A334632. (End)

A134370 a(n) = ((2n+1)!)^(n+2).

Original entry on oeis.org

1, 216, 207360000, 3252016064102400000, 2283380023591730815784976384000000, 161469323688736156802100136913438716723200000000000000, 2260697901194635682690248130915498742378267539496871953338204160000000000000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Crossrefs

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134371, A134374, A134375.

Programs

  • Mathematica
    Table[((2n+1)!)^(n + 2), {n, 0, 10}]

Formula

a(n) ~ 2^(2*(n+1)*(n+2)) * exp(13/24 - 2*n*(n+2)) * n^((n+2)*(4*n+3)/2) * Pi^(n/2 + 1). - Vaclav Kotesovec, Oct 26 2017

Extensions

Typo in a(6) corrected by Georg Fischer, Apr 10 2024

A134373 a(n) = ((2n)!)^3.

Original entry on oeis.org

1, 8, 13824, 373248000, 65548320768000, 47784725839872000000, 109903340320478724096000000, 662559760549147780765974528000000, 9159226129831418921308831875072000000000, 262435789155225791087396177124997988352000000000
Offset: 0

Views

Author

Artur Jasinski, Oct 22 2007

Keywords

Links

Crossrefs

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134373, A134374, A134375.

Programs

  • Mathematica
    Table[((2n)!)^(3), {n, 0, 10}]
((2*Range[0, 10])!)^3 (* Harvey P. Dale, Jul 25 2016 *)
  • Sage
    [factorial(2*n)**3 for n in range(0,9)] # Stefano Spezia, Apr 22 2025

Extensions

Definition corrected by Harvey P. Dale, Jul 25 2016

A198481 Square root of the largest square dividing ((2n-1)!)^(2n-3).

Original entry on oeis.org

1, 1, 240, 304819200, 3440500260470784000, 1827912356210202139164672000000000, 13482302715547740229948201750717130814259200000000000
Offset: 1

Views

Author

Artur Jasinski, Oct 25 2011

Keywords

Comments

For the complementary squarefree parts see A197880.

Crossrefs

Cf. A134367, A197880.

Programs

  • Maple
    A000188 := proc(n)
        a := 1 ;
        for pf in ifactors(n)[2] do
                p := op(1,pf) ;
                e := op(2,pf) ;
                a := a*p^(floor(e/2)) ;
        end do:
        a ;
end proc:
A198481 := proc(n)
        A000188( A134367(2*n-1)) ;
end proc:
seq(A198481(n),n=1..10) ; # R. J. Mathar, Oct 25 2011
  • Mathematica
    aa = {}; data = Table[kk = Sqrt[(n!)^(n - 2)], {n, 1, 100, 2}]; sp = data /. Sqrt[_] -> 1; sfp = data/sp; sp
Sqrt[#]&/@Table[Max[Select[Divisors[((2n-1)!)^(2n-3)],IntegerQ[Sqrt[#]]&]],{n,7}] (* Harvey P. Dale, May 24 2024 *)

Formula

a(n) = A000188(A134367(2*n-1)). - R. J. Mathar, Oct 25 2011
Showing 1-10 of 12 results. Next

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