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

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

Offset: 0

Author

Seiichi Manyama, May 20 2025

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A132393, A242676, A383881, A383882.

Cf. A187646, A384129.

Programs

Magma

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

Mathematica

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));

Formula

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