cp's OEIS Frontend

E.g.f.: A(x) = (1 - 2*x - x^2)^(-1/2). - Paul Boddington, Nov 17 2003

a(n) = n!/2^n*A006139(n) = n!*Sum_{k=floor(n/2)..n} 2^(k-n)*C(n, k)*C(k, n-k). Sum_{n>=0} a(n)*x^n/n!^2 = exp(x)*BesselI(0, sqrt(2)*x). a(n) is the central coefficient of n!*(1+x+x^2/2)^n. - Vladeta Jovovic, Mar 22 2004

From Peter Bala, Aug 25 2011: (Start)

The function B(x) := int {t=0..x} A(t), obtained by integrating the generating function A(x), satisfies the autonomous differential equation d/dx(B(x)) = 1/(cos(B(x))-sin(B(x))). Compare with A190392.

Thus B(x), and hence A(x), can be found by inverting the function int {t=0..x} (cos(t)-sin(t)). By applying [Dominici, Theorem 4.1] the result can be expressed as

A(x) = 1 + sum {n>=1} D^n[1/(cos(t)-sin(t))](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n) = D^n[1/(cos(t)-sin(t))](0). (End)

E.g.f. at offset 1: Series_Reversion(cos(x) + sin(x) - 1). - Paul D. Hanna, Aug 08 2012

a(n) ~ (1+sqrt(2))^(n+1/2) * n^n / (2^(1/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017