A384129 Number of permutations of 3*n objects with exactly 2*n cycles.
1, 3, 85, 4536, 357423, 37312275, 4853222764, 756111184500, 137272511800831, 28460103232088385, 6634460278534540725, 1717750737160208150400, 489078062391738506912340, 151874660255802127280374140, 51082995429153110239690350120, 18500755859447038660174079965500
Offset: 0
Programs
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PARI
a(n) = abs(stirling(3*n, 2*n, 1));
Formula
a(n) = A132393(3*n,2*n) = |Stirling1(3*n,2*n)|.
a(n) = (3*n)! * [x^(3*n)] log(1 - x)^(2*n) / (2*n)!.
a(n) ~ 3^(4*n - 1/2) * w^(3*n) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 2^(2*n + 1/2) * exp(n) * (3*w-2)^n), where w = -LambertW(-1, -2*exp(-2/3)/3) = 1.4293552275170056487... - Vaclav Kotesovec, May 23 2025