A100732 -id:A100732 - cp's OEIS frontend

A100733 a(n) = (4*n)!.

1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000

Offset: 0

Ralf Stephan, Dec 08 2004

From Karol A. Penson, Jun 11 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*exp(-x^(1/4))/x^(3/4) on the positive axis:
a(n) = Integral_{x=0..oo} x^n*W(x) dx = Integral_{x=0..oo} x^n*(1/4)*exp(-x^(1/4))/x^(3/4) dx, n >= 0.
This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.
As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
This is indeed the case as by construction the following "doubly" infinite family:
V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),
with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100732, A100734, A268505, A332890.

[Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(4*Range[0,20])! (* Harvey P. Dale, Oct 03 2014 *)

E.g.f.: 1/(1-x^4).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021

More terms from Harvey P. Dale, Oct 03 2014

A268504 a(n) = Product_{k=0..n} (3*k)!.

Original entry on oeis.org

1, 6, 4320, 1567641600, 750902834626560000, 981936389699695364014080000000, 6286723722110812136775527266768650240000000000, 321194638135877430211257700556824829511701622266265600000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100732. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..22

Cf. A000178, A098694, A100732, A268196, A268505, A268506, A271946, A271947.

Mathematica
```
Table[Product[(3*k)!,{k,0,n}],{n,0,10}]
```

{a(n) = prod(k=1, n, (3*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36) * 2^(n/2 + 1/3) * Pi^(n/2 + 1/3) * n^(3*n^2/2 + 2*n + 19/36) / (A^(1/3) * exp(9*n^2/4 + 2*n - 1/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A100734 a(n) = (5*n)!.

Original entry on oeis.org

1, 120, 3628800, 1307674368000, 2432902008176640000, 15511210043330985984000000, 265252859812191058636308480000000, 10333147966386144929666651337523200000000

Offset: 0

Ralf Stephan, Dec 08 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A000142, A010050, A100732, A100733, A268506, A269296.

[Factorial(5*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(5*Range[0,10])! (* Harvey P. Dale, May 06 2015 *)

E.g.f.: 1/(1-x^5).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*5^(5*n + 1/2)*n^(5*n + 1/2)/exp(5*n).
Sum_{n>=0} 1/a(n) = A269296. (End)

A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 841, 5251, 20497, 423865, 3780721, 20292031, 559501801, 6487717237, 44317795705, 1527439916731, 21798729916321, 180816606476401, 7478345832314977, 126737815733490295, 1236785588298582841, 59677199741873516461, 1171057417377450325801

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000522, A087208, A100732, A158757, A330045.

[n le 3 select 1 else 1 + 6*Binomial(n-1,3)*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 05 2021

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]

[sum(factorial(3*k)*binomial(n, 3*k) for k in (0..n//3)) for n in (0..40)] # G. C. Greubel, Dec 05 2021

G.f.: Sum_{k>=0} (3*k)! * x^(3*k) / (1 - x)^(3*k + 1).
a(0) = a(1) = a(2) = 1; a(n) = n * (n - 1) * (n - 2) * a(n - 3) + 1.
a(n) = Sum_{k=0..floor(n/3)} n! / (n - 3*k)!.
a(n) ~ n! * (exp(1)/3 + 2*cos(sqrt(3)/2 - 2*Pi*n/3) / (3*exp(1/2))). - Vaclav Kotesovec, Apr 18 2020
a(n) = A158757(n, 2*n). - G. C. Greubel, Dec 05 2021

A195390 a(n) = (6*n)!.

Original entry on oeis.org

1, 720, 479001600, 6402373705728000, 620448401733239439360000, 265252859812191058636308480000000, 371993326789901217467999448150835200000000, 1405006117752879898543142606244511569936384000000000, 12413915592536072670862289047373375038521486354677760000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..75

Cf. A000142, A008588, A010050, A100732, A100733, A100734, A271946, A332892.

Magma
```
[Factorial(6*n): n in [0..10]];
```

(6*Range[0,10])! (* Harvey P. Dale, Dec 16 2013 *)

From Amiram Eldar, Apr 03 2021: (Start)
a(n) = A000142(A008588(n)).
Sum_{n>=0} 1/a(n) = A332892. (End)

A195391 a(n) = (7*n)!.

Original entry on oeis.org

1, 5040, 87178291200, 51090942171709440000, 304888344611713860501504000000, 10333147966386144929666651337523200000000, 1405006117752879898543142606244511569936384000000000, 608281864034267560872252163321295376887552831379210240000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A271947.

Magma
```
[Factorial(7*n): n in [0..10]];
```

(7*Range[0, 8])! (* Paolo Xausa, Aug 12 2025 *)

a(n)=(7*n)! \\ Charles R Greathouse IV, Dec 27 2011

A143084 Triangle read by rows: T(n,m) = (n + m)!.

Original entry on oeis.org

1, 1, 2, 2, 6, 24, 6, 24, 120, 720, 24, 120, 720, 5040, 40320, 120, 720, 5040, 40320, 362880, 3628800, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 16 2008

			Triangle begins:
    1;
    1,    2;
    2,    6,    24;
    6,   24,   120,    720;
   24,  120,   720,   5040,   40320;
  120,  720,  5040,  40320,  362880,  3628800;
  720, 5040, 40320, 362880, 3628800, 39916800, 479001600;
  ...

Jinyuan Wang, Rows n = 0..50 of triangle, flattened

Column m=0 gives A000142.
T(2n,n) gives A100732.
Main diagonal gives A010050.
Row sums give A374574.

t[n_, m_] := (n + m)!; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

T(n,m) = (n + m)!.

Offset changed to 0 by Jinyuan Wang, Dec 19 2020

A195392 a(n) = (8*n)!.

Original entry on oeis.org

1, 40320, 20922789888000, 620448401733239439360000, 263130836933693530167218012160000000, 815915283247897734345611269596115894272000000000, 12413915592536072670862289047373375038521486354677760000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..55

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391

Magma
```
[Factorial(8*n): n in [0..10]];
```

(8*Range[0, 8])! (* Paolo Xausa, Aug 12 2025 *)

A065961 a(n) = (3n - 1)!n/2.

Original entry on oeis.org

1, 120, 60480, 79833600, 217945728000, 1067062284288000, 8515157028618240000, 103408066955539906560000, 1814811575069725360128000000, 44208809968698509772718080000000, 1447219603135314415919699066880000000, 61998887798316869577999908025139200000000

Offset: 1

Len Smiley and George E. Antoniou, Dec 08 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..60

Cf. A100732.

[ Factorial(3*n)/6: n in [1..62] ]; // Vincenzo Librandi, Apr 24 2011

(3*Range[15])!/6 (* Paolo Xausa, Feb 16 2024 *)

{ for (n=1, 60, a=(3*n - 1)!*n/2; write("b065961.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 04 2009

a(n) = (3*n)! / 6.
a(n) - 3*n*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015
From Amiram Eldar, Feb 17 2024: (Start)
Sum_{n>=1} 1/a(n) = 2*e - 6 + 4*cos(sqrt(3)/2)/sqrt(e).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6 - 2/e - 4*sqrt(e)*cos(sqrt(3)/2). (End)

A195393 a(n) = (9*n)!.

Original entry on oeis.org

1, 362880, 6402373705728000, 10888869450418352160768000000, 371993326789901217467999448150835200000000, 119622220865480194561963161495657715064383733760000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391, A195392.

Magma
```
[Factorial(9*n): n in [0..10]];
```

(9Range[0,10])! (* Harvey P. Dale, Jan 25 2023 *)

cp's OEIS Frontend

A100733 a(n) = (4*n)!.

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A268504 a(n) = Product_{k=0..n} (3*k)!.

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A100734 a(n) = (5*n)!.

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A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

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Sage

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A195390 a(n) = (6*n)!.

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Magma

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A195391 a(n) = (7*n)!.

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Magma

Mathematica

PARI

A143084 Triangle read by rows: T(n,m) = (n + m)!.

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Examples

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Programs

Mathematica

Formula

Extensions

A195392 a(n) = (8*n)!.

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Keywords

A065961 a(n) = (3n - 1)!n/2.