A383882 -id:A383882 - cp's OEIS frontend

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

1, 6, 266, 22275, 2757118, 452329200, 92484925445, 22653141490980, 6466506598695390, 2108114165258886708, 772778072287000494520, 314641228029527540596455, 140880584836935832288402135, 68799366730032076856334789900, 36392216443342587869022660451080, 20728132932716479897744043460870000

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

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

Cf. A007820, A348084, A383882.

Cf. A217913.

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

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 3*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[4*n, 3*n], {n, 0, 15}]
Table[SeriesCoefficient[(-1)^n/(Pochhammer[1 - 1/x, 3*n]*x^(3*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(4*n,3*n).

a(n) ~ (-1)^(3*n) * 4^(4*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 3^(3*n + 1/2) * w^(3*n) * (4/3 + w)^n), where w = LambertW(-4/(3*exp(4/3))).

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

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

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

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Author

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Programs

Magma

Mathematica

PARI

Formula

cp's OEIS Frontend

A383881 a(n) = [x^n] Product_{k=1..3*n} 1/(1 - k*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

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

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

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