A100733 -id:A100733 - cp's OEIS frontend

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

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)

A268505 a(n) = Product_{k=0..n} (4*k)!.

Original entry on oeis.org

1, 24, 967680, 463520268288000, 9698137182219213471744000000, 23594617426193665303453830729600860160000000000, 14639242671589099207353038379393488170313478620292159897600000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100733. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..19

Cf. A000178, A098694, A100733, A268504, A268506, A271946, A271947.

Mathematica
```
Table[Product[(4*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (4*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/4)^(1/2) * 2^(5/6 + 11*n/2 + 4*n^2) * exp(1/48 - 5*n/2 - 3*n^2) * n^(29/48 + 5*n/2 + 2*n^2) * Pi^(1/4 + n/2) / A^(1/4), where A = A074962 is the Glaisher-Kinkelin constant.

a(n) = A^(15/4) * sqrt(Gamma(1/4)) * exp(-5/16) * 2^(4*n^2 + 7*n/2 - 5/12) * BarnesG(n + 5/4) * BarnesG(n + 3/2) * BarnesG(n + 7/4) * BarnesG(n+2) / Pi^(3*n/2 + 1). - Vaclav Kotesovec, Apr 23 2024

A100732 a(n) = (3*n)!.

Original entry on oeis.org

1, 6, 720, 362880, 479001600, 1307674368000, 6402373705728000, 51090942171709440000, 620448401733239439360000, 10888869450418352160768000000, 265252859812191058636308480000000

Offset: 0

Ralf Stephan, Dec 08 2004

a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 12 2011

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

Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.

a100732 = a000142 . a008585  -- Reinhard Zumkeller, Feb 19 2013

[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011

Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* John M. Campbell, Jul 12 2011 *)
(3Range[0,10])! (* Harvey P. Dale, Sep 23 2011 *)

[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016

a(n) = A000142(A008585(n)).
E.g.f.: 1/(1-x^3).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).
Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - Amiram Eldar, Feb 14 2021
a(n) = A143084(2n,n). - Alois P. Heinz, Jul 12 2024

A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, 1819441, 6660721, 498971881, 6278929801, 43710250585, 218205219961, 21795091762081, 358652470233121, 3210080802962401, 20298322381652065, 2534333270094778681, 51516840824285500441, 563561785768079119561

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000522, A087208, A100733, A330044, A334157.

Outer diagonal of A158777.

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]

G.f.: Sum_{k>=0} (4*k)! * x^(4*k) / (1 - x)^(4*k + 1).
a(0) = a(1) = a(2) = a(3) = 1; a(n) = n*(n - 1)*(n - 2)*(n - 3)*a(n - 4) + 1.
a(n) = Sum_{k=0..floor(n/4)} n! / (n - 4*k)!.
a(n) ~ n! * (2*cos(Pi*n/2 - 1) + exp(1) + (-1)^n*exp(-1))/4. - Vaclav Kotesovec, Apr 18 2020

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)

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]

a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...

a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.

Original entry on oeis.org

1, 3, 315, 155925, 212837625, 618718975875, 3287253918823875, 28845653137679503125, 388983632561608099640625, 7637693625347175036443671875, 209402646126143497974176151796875, 7752714167528210725497923667975703125, 377130780679409810741846496828678078515625

Offset: 0

Reinhard Zumkeller, Apr 25 2008

nonn

From Karol A. Penson, Oct 05 2009: (Start)
Integral representation as n-th moment of a positive function on a positive semi-axis (solution of the Stieltjes moment problem), in Maple notation:
a(n)=int(x^n*((1/4)*sqrt(2)*(Pi^(3/2)*2^(1/4)*hypergeom([], [1/2, 3/4], -(1/32)*x)*sqrt(x)-2*Pi*hypergeom([], [3/4, 5/4], -(1/32)*x)*GAMMA(3/4)*x^(3/4)+sqrt(Pi)*GAMMA(3/4)^2*2^(1/4)*hypergeom([], [5/4, 3/2],-(1/32)*x)*x)/(Pi^(3/2)*GAMMA(3/4)*x^(5/4))), x=0..infinity), n=0,1... .
This solution may not be unique. (End)

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Appendix: Problem 203.1, p164.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Tournament
Index entries for sequences related to tornaments.

Cf. A008299, A000142, A100733, A001018.

a(n)={(4*n)!/(n!*8^n)} \\ Andrew Howroyd, Jan 07 2020

a(n) = (4*n)!/(n!*8^n).
a(n) = A001147(n)*A001147(2*n).
a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - Reinhard Zumkeller, Apr 28 2008

Terms a(11) and beyond from Andrew Howroyd, Jan 07 2020

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

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 *)

A381162 a(n) = (8n)!/((n!)^4(4*n)!).

Original entry on oeis.org

1, 1680, 32432400, 999456057600, 37905932634570000, 1617318175088527591680, 74451445170005824874553600, 3614146643656788883257309696000, 182458061523203642337177421198794000, 9493111901274733909567003010522405280000, 505860213332178847817809654781948251947782400

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 7.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See pp. 14-15.

Cf. A100733, A134375, A195392.

Cf. A381161, A381163, A381164, A381165.

a[n_]:=(8n)!/((n!)^4*(4n)!); Array[a,11,0]

G.f.: hypergeom([1/8, 3/8, 5/8, 7/8], [1, 1, 1], 2^16*x).

a(n) ~ 2^(16*n - 3/2) / (Pi^2*n^2). - Vaclav Kotesovec, May 29 2025

cp's OEIS Frontend

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

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A268505 a(n) = Product_{k=0..n} (4*k)!.

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A100732 a(n) = (3*n)!.

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Haskell

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Sage

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A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

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Mathematica

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

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Magma

Mathematica

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A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

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Mathematica

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A139541 There are 4*n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4*n)!/(n!*8^n) different ways.

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PARI

Formula

Extensions

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

Views

Author

Keywords

Links

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.

A381162 a(n) = (8n)!/((n!)^4(4*n)!).