A346966 Expansion of e.g.f. -log( 1 - log(1 - x)^2 / 2 ).

1, 3, 14, 80, 559, 4599, 43665, 470196, 5666586, 75600690, 1106587008, 17636532264, 304092954138, 5640892517610, 112029356591862, 2371963759970352, 53338181764577304, 1269586152655203672, 31891196481381667008, 843109673024218773600, 23400930987874505081160

Offset: 2

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

• nonn

Crossrefs

Cf. A000254, A003713, A346921, A346944.

Programs

Mathematica

nmax = 22; CoefficientList[Series[-Log[1 - Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = Abs[StirlingS1[n, 2]] + (1/n) Sum[Binomial[n, k] Abs[StirlingS1[n - k, 2]] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 22}]

Formula

a(n) = |Stirling1(n,2)| + (1/n) * Sum_{k=1..n-1} binomial(n,k) * |Stirling1(n-k,2)| * k * a(k).

a(n) ~ (n-1)! / (1 - exp(-sqrt(2)))^n. - Vaclav Kotesovec, Jun 04 2022

a(n) = Sum_{k=1..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(k * 2^k). - Seiichi Manyama, Jan 23 2025