A385369 Expansion of e.g.f. x + sqrt(x^2 + 1).

1, 1, 1, 0, -3, 0, 45, 0, -1575, 0, 99225, 0, -9823275, 0, 1404728325, 0, -273922023375, 0, 69850115960625, 0, -22561587455281875, 0, 9002073394657468125, 0, -4348001449619557104375, 0, 2500100833531245335015625, 0, -1687568062633590601135546875, 0

Offset: 0

Author

Seiichi Manyama, Jun 26 2025

Keywords

• sign
• easy

Crossrefs

Cf. A006228, A177698, A385343.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(x+sqrt(x^2+1)))

Formula

E.g.f.: exp(arcsinh(x)).

E.g.f. A(x) satisfies A(x) = 1/A(-x).

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

a(n) = A177698(n-1) for n > 1.

a(2*n+1) = 0 for n > 0.

a(n) = 2^n * n! * binomial((n+1)/2,n)/(n+1).