A217913 -id:A217913 - cp's OEIS frontend

A237993 a(n) = |Stirling1(3*n,n)|.

1, 2, 274, 118124, 105258076, 159721605680, 369012649234384, 1206647803780373360, 5304713715525445812976, 30180059720580991603896800, 215760462268683520394805979744, 1893448925578239663637174767335168, 20012008248418194052035539503977759232

Offset: 0

Vaclav Kotesovec, May 20 2014

Cf. A132393, A187646, A242676, A217913.

Maple
```
seq(abs(Stirling1(3*n,n)), n=0..20);
```
Mathematica
```
Table[Abs[StirlingS1[3*n, n]],{n,0,20}]
```

a(n) ~ n^(2*n) * c^(3*n) * 3^(5*n) / (sqrt(6*Pi*(c-1)*n) * exp(2*n) * (3*c-1)^(2*n)), where c = -LambertW(-1,-exp(-1/3)/3) = 2.237147027773716818...
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(3*n,n).
a(n) = (3*n)! * [x^(3*n)] (-log(1 - x))^n / n!. (End)

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

Original entry on oeis.org

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))).

cp's OEIS Frontend

A237993 a(n) = |Stirling1(3*n,n)|.

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Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A237993 a(n) = |Stirling1(3*n,n)|.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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