If offset is 0, a(n) = Sum_{i=0..n} binomial(n+1, i+1)*Bell(i) [cf.
A000110].
Recursively defined continued fractions:
G.f.: G(0)/(1-x) where G(k) = 1 - 2*x*(k + 1)/((2*k + 1)*(2*x*k + x - 1) - x*(2*k + 1)*(2*k + 3)*(2*x*k + x - 1)/(x*(2*k + 3) - 2*(k + 1)*(2*x*k + 2*x - 1)/G(k+1))).
G.f.: (G(0) - 1)/(1 - x) where G(k) = 1 + (1 - x)/(1 - x*(k + 1))/(1 - x/(x + (1 -x)/G(k+1))).
G.f.: (S - 1)/(1 - x), where S = (1/(1 - x)) * Sum_{k>=0} ((1 + (1 - x)/(1 - x -x*k))*x^k / Product_{i=1..k-1} (1 - x - x*i)).
G.f.: ((G(0) - 2)/(2*x - 1) - 1)/(1 - x)/x where G(k) = 2 - 1/(1 - k*x)/(1 - x/(x - 1/G(k+1))).
G.f.: 1/(G(0) - x)/(1 - x), where G(k) = 1 - x*(k + 1)/(1 - x/G(k+1)). (End)
a(n) = (1/e)*Sum_{k>=1} (k^n - 1)/((k - 1)*(k - 1)!). -
Joseph Wheat, Jul 16 2024
