A384129 -id:A384129 - cp's OEIS frontend

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

1, 10, 750, 106470, 22350954, 6220194750, 2157580085700, 896587036640680, 434225240080346858, 240175986308550372366, 149377949042637543000150, 103192471874508023383125750, 78394850841083734162487127720, 64957213308036504429927388238088, 58298851680969051596827194829579744

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

In general, for m>=1, Stirling2((m+1)*n, m*n) ~ (-1)^(m*n) * (m+1)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w(m))) * exp(n) * m^(m*n + 1/2) * w(m)^(m*n) * (1 + 1/m + w(m))^n), where w(m) = LambertW(-(1 + 1/m)/exp(1 + 1/m)).

Cf. A007820 (m=1), A348084 (m=2), A383881 (m=3).
Cf. A217913.
Cf. A187646, A384129, A384130.

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 4*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[5*n, 4*n], {n, 0, 15}]
Table[SeriesCoefficient[1/(Pochhammer[1 - 1/x, 4*n]*x^(4*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(5*n,4*n).

a(n) ~ 5^(5*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 4^(4*n + 1/2) * w^(4*n) * (5/4 + w)^n), where w = LambertW(-5/(4*exp(5/4))).

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

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

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

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cp's OEIS Frontend

A383882 a(n) = [x^n] Product_{k=1..4*n} 1/(1 - k*x).

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Author

Keywords

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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

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

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