A384501 a(n) = Sum_{k=0..n} abs(Stirling1(n,k)) * Stirling2(n,n-k).

1, 0, 1, 9, 119, 2025, 42510, 1062761, 30854159, 1020615912, 37900765365, 1561459425955, 70682817696436, 3487456195458027, 186281997929231659, 10709829446929099865, 659427284782849503663, 43293574636994934145044, 3019108475859713906967738, 222868205832269470083471366

Offset: 0

Author

Vaclav Kotesovec, May 31 2025

Keywords

• nonn

Crossrefs

Cf. A047793, A187655, A187656.

Programs

Mathematica

Table[Sum[Abs[StirlingS1[n, k]]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]
Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[n, n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} abs(Stirling1(n,n-k)) * Stirling2(n,k).

a(n) ~ c * ((-r - 1/((1-r)*LambertW(exp(1/(r-1))/(r-1)))) / (1 + (1-r)*LambertW(exp(1/(r-1))/(r-1))))^n * n^(n - 1/2) / exp(n), where r = 0.412059483521755003540032983286575579547027818844750... is the root of the equation (1-r)^2 * (1 + LambertW(-1, -exp(-r)*r)/r) = (1-r) + 1/LambertW(exp(1/(r-1))/(r-1)) and c = 0.21367572159147979376975234273...