A242676 -id:A242676 - cp's OEIS frontend

A187646 (Signless) Central Stirling numbers of the first kind s(2n,n).

1, 1, 11, 225, 6769, 269325, 13339535, 790943153, 54631129553, 4308105301929, 381922055502195, 37600535086859745, 4070384057007569521, 480544558742733545125, 61445535102359115635655, 8459574446076318147830625, 1247677142707273537964543265, 196258640868140652967646352465

Offset: 0

Emanuele Munarini, Mar 12 2011

Number of permutations with n cycles on a set of size 2n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A008275, A007820, A129505, A132393, A187235, A237993, A242676, A285862.

Maple
```
seq(abs(Stirling1(2*n,n)), n=0..20);
```

Table[Abs[StirlingS1[2n, n]], {n, 0, 12}]
N[1 + 1/(2 LambertW[-1, -Exp[-1/2]/2]), 50] (* The constant z in the asymptotic - Vladimir Reshetnikov, Oct 08 2016 *)

makelist(abs(stirling1(2*n,n)),n,0,12);

for(n=0,50, print1(abs(stirling(2*n, n, 1)), ", ")) \\ G. C. Greubel, Nov 09 2017

Asymptotic: a(n) ~ (2*n/(e*z*(1-z)))^n*sqrt((1-z)/(2*Pi*n*(2z-1))), where z=0.715331862959... is a root of the equation z = 2*(z-1)*log(1-z). - Vaclav Kotesovec, May 30 2011
Equivalent: a(n) ~ n!*(2*r^2/(r-1))^n/(2*Pi*n*sqrt(r-2)), where r=A226278. - Natalia L. Skirrow, Jul 13 2025
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(2*n,n).
a(n) = (2*n)! * [x^(2*n)] (-log(1 - x))^n / n!. (End)

A237993 a(n) = |Stirling1(3*n,n)|.

Original entry on oeis.org

1, 2, 274, 118124, 105258076, 159721605680, 369012649234384, 1206647803780373360, 5304713715525445812976, 30180059720580991603896800, 215760462268683520394805979744, 1893448925578239663637174767335168, 20012008248418194052035539503977759232

Offset: 0

Vaclav Kotesovec, May 20 2014

Cf. A132393, A187646, A242676, A217913.

Maple
```
seq(abs(Stirling1(3*n,n)), n=0..20);
```
Mathematica
```
Table[Abs[StirlingS1[3*n, n]],{n,0,20}]
```

a(n) ~ n^(2*n) * c^(3*n) * 3^(5*n) / (sqrt(6*Pi*(c-1)*n) * exp(2*n) * (3*c-1)^(2*n)), where c = -LambertW(-1,-exp(-1/3)/3) = 2.237147027773716818...
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(3*n,n).
a(n) = (3*n)! * [x^(3*n)] (-log(1 - x))^n / n!. (End)

A384130 Number of permutations of 4n objects with exactly 3n cycles.

Original entry on oeis.org

1, 6, 322, 32670, 4899622, 973941900, 241276443496, 71603372991150, 24764667228756390, 9781650150525639540, 4344363139637533397580, 2143082171052546774398348, 1162585907585797437278546956, 687872810620417599693839111880, 440840269604491448260396623711300

Offset: 0

Seiichi Manyama, May 20 2025

In general, for m>=1, abs(Stirling1((m+1)*n, m*n)) ~ (m+1)^((m+2)*n - 1/2) * w(m)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(w(m)-1)) * exp(n) * m^(m*n) * ((m+1)*w(m) - m)^n), where w(m) = -LambertW(-1, -m*exp(-m/(m+1))/(m+1)). - Vaclav Kotesovec, May 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A132393, A242676, A383881, A383882.

Cf. A187646, A384129.

[&+[Abs(StirlingFirst(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

a[n_]:=Abs[StirlingS1[4 n,3 n]] Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 21 2025 *)

PARI
```
a(n) = abs(stirling(4*n, 3*n, 1));
```

a(n) = A132393(4*n,3*n) = |Stirling1(4*n,3*n)|.
a(n) = (4*n)! * [x^(4*n)] (-log(1 - x))^(3*n) / (3*n)!.
a(n) ~ 2^(10*n - 3/2) * n^(n - 1/2) * w^(4*n) / (sqrt(Pi*(w-1)) * 3^(3*n) * exp(n) * (4*w-3)^n), where w = -LambertW(-1, -3*exp(-3/4)/4) = 1.3002007416590685881... - Vaclav Kotesovec, May 23 2025

cp's OEIS Frontend

A187646 (Signless) Central Stirling numbers of the first kind s(2n,n).

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A237993 a(n) = |Stirling1(3*n,n)|.

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A384130 Number of permutations of 4n objects with exactly 3n cycles.

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cp's OEIS Frontend

A187646 (Signless) Central Stirling numbers of the first kind s(2n,n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A237993 a(n) = |Stirling1(3*n,n)|.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A384130 Number of permutations of 4*n objects with exactly 3*n cycles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A384130 Number of permutations of 4n objects with exactly 3n cycles.