cp's OEIS Frontend

E.g.f. A(x) satisfies A(x) = exp( arcsinh(x) * A(x) ).

E.g.f. A(x) satisfies A(x) = ( x + sqrt(x^2 + 1) )^A(x).

a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

From Vaclav Kotesovec, Jun 28 2025: (Start)

a(n) ~ 2^n * exp((exp(-1) - 1)*n + 3/2) * n^(n-1) / (sqrt(1 + exp(2*exp(-1))) * (exp(2*exp(-1)) - 1)^(n - 1/2)).

Equivalently, a(n) ~ n^(n-1) / (sqrt(cosh(exp(-1))) * sinh(exp(-1))^(n - 1/2) * exp(n - 3/2)). (End)

E.g.f. A(x) satisfies A(x) = exp( arcsin(x*A(x)) ).

a(n) = Sum_{k=0..n} (n+1)^(k-1) * A385343(n,k).

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A385343(n,k).

a(n) ~ s*(1 - r^2*s^2)^(3/4) * n^(n-1) / (sqrt(r^2*s^2*(2 + r*sqrt(1 - r^2*s^2) - r^2*s^2) - 1) * exp(n) * r^(n - 1/2)), where r = 0.4947196925654744939290429342422921705036054462455... and s = 1.929162378596122962197524561455700427559144822670... are the roots of the system of equations exp(arcsin(r*s)/s) = s, r*s/sqrt(1 - r^2*s^2) - arcsin(r*s) = s. - Vaclav Kotesovec, Jul 05 2025