A242676 a(n) = |Stirling1(4*n,n)|.

1, 6, 13068, 150917976, 5056995703824, 371384787345228000, 50779532534302850198976, 11616723683566425573507775872, 4123257155075936045020928754053376, 2146734309994687055429549444238169536000, 1569808063009967047226374755685187772671339520

Offset: 0

Author

Vaclav Kotesovec, May 20 2014

Keywords

• nonn
• easy

Comments

Generally, for p>=2 is Abs(StirlingS1(p*n,n)) asymptotic to n^((p-1)*n) * c^(p*n) * p^((2*p-1)*n) / (sqrt(2*Pi*p*(c-1)*n) * exp((p-1)*n) * (c*p-1)^((p-1)*n)), where c = -LambertW(-1,-exp(-1/p)/p).

Crossrefs

Cf. A187646, A237993, A217914.

Programs

Maple

seq(abs(Stirling1(4*n,n)), n=0..20);

Mathematica

Table[Abs[StirlingS1[4*n, n]],{n,0,20}]

Formula

a(n) ~ n^(3*n) * c^(4*n) * 2^(14*n-1) / (sqrt(2*Pi*(c-1)*n) * exp(3*n) * (4*c-1)^(3*n)), where c = -LambertW(-1,-exp(-1/4)/4) = 2.58666298226305388118285...

From Seiichi Manyama, May 20 2025: (Start)

a(n) = A132393(4*n,n).

a(n) = (4*n)! * [x^(4*n)] (-log(1 - x))^n / n!. (End)