A217914 -id:A217914 - cp's OEIS frontend

A218143 a(n) = Stirling2(n*(n+1)/2, n).

1, 1, 3, 90, 34105, 210766920, 26585679462804, 82892803728383735268, 7529580759157036060608585183, 22982258052528294182955639980819773510, 2672446997421818663856559987803834697952486978300, 13239043631590111512460321918828937597837325561187113535696980

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 90*x^3 + 34105*x^4 + 210766920*x^5 + 26585679462804*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 0..30

Cf. A008277, A218141, A218142, A007820, A217913, A217914, A217915.

Table[StirlingS2[n*(n+1)/2, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n*(n+1)/2,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n*(n-1)/2))), n*(n-1)/2)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n*(n+1)/2, n)}
for(n=0, 15, print1(a(n), ", "))

a(n) = [x^(n*(n-1)/2)] 1 / Product_{k=1..n} (1-k*x).

a(n) ~ n^(n*(n+1)/2)/n!. - Vaclav Kotesovec, May 11 2014

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

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 20 2014

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).

Cf. A187646, A237993, A217914.

Maple
```
seq(abs(Stirling1(4*n,n)), n=0..20);
```
Mathematica
```
Table[Abs[StirlingS1[4*n, n]],{n,0,20}]
```

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)

cp's OEIS Frontend

A218143 a(n) = Stirling2(n*(n+1)/2, n).

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