This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384130 #26 May 23 2025 06:18:23
%S A384130 1,6,322,32670,4899622,973941900,241276443496,71603372991150,
%T A384130 24764667228756390,9781650150525639540,4344363139637533397580,
%U A384130 2143082171052546774398348,1162585907585797437278546956,687872810620417599693839111880,440840269604491448260396623711300
%N A384130 Number of permutations of 4*n objects with exactly 3*n cycles.
%C A384130 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
%H A384130 Vincenzo Librandi, <a href="/A384130/b384130.txt">Table of n, a(n) for n = 0..200</a>
%F A384130 a(n) = A132393(4*n,3*n) = |Stirling1(4*n,3*n)|.
%F A384130 a(n) = (4*n)! * [x^(4*n)] (-log(1 - x))^(3*n) / (3*n)!.
%F A384130 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
%t A384130 a[n_]:=Abs[StirlingS1[4 n,3 n]] Table[a[n],{n,0,15}] (* _Vincenzo Librandi_, May 21 2025 *)
%o A384130 (PARI) a(n) = abs(stirling(4*n, 3*n, 1));
%o A384130 (Magma) [&+[Abs(StirlingFirst(4*n, 3*n))]: n in [0..15]]; // _Vincenzo Librandi_, May 21 2025
%Y A384130 Cf. A132393, A242676, A383881, A383882.
%Y A384130 Cf. A187646, A384129.
%K A384130 nonn,easy
%O A384130 0,2
%A A384130 _Seiichi Manyama_, May 20 2025