A342166 Product of first n Fubini numbers.

1, 1, 3, 39, 2925, 1582425, 7410496275, 350464600333575, 191295845123076910125, 1355763582602823185129417625, 138623522325287867599380791765497875, 224935042709004795568466587349227029537282375, 6318777956744220129890735589019782971247629409914638125

Offset: 0

Author

Vaclav Kotesovec, Mar 03 2021

Keywords

• nonn

Links

• Wikipedia, Ordered Bell number

Crossrefs

Cf. A000670, A003046, A003266, A070825, A294373, A342170.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*g(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Mathematica

Table[Product[Sum[j!*StirlingS2[k, j], {j, 0, k}], {k, 1, n}], {n, 0, 12}]
Table[Product[PolyLog[-k, 1/2]/2, {k, 1, n}], {n, 0, 12}]
FoldList[Times, 1, Table[PolyLog[-n, 1/2]/2, {n, 1, 12}]]

Formula

a(n) = Product_{k=1..n} A000670(k).

a(n) ~ c * BarnesG(n+2) / (2^n * log(2)^(n*(n+3)/2)), where c = 0.960303470666951851619546415046950178638511457142008903473074598398282549...

a(n) ~ c * Pi^((n+1)/2) * n^(n^2/2 + n + 5/12) / (A * 2^((n-1)/2) * exp(3*n^2/4 + n - 1/12) * log(2)^(n*(n+3)/2)), where A is the Glaisher-Kinkelin constant A074962.