A352121 Expansion of e.g.f. sqrt(2 - exp(-2*x)).

1, 1, -3, 13, -87, 841, -10683, 167413, -3113967, 66991441, -1635760563, 44683635613, -1350018280647, 44694643670041, -1608962582321643, 62572776778020613, -2614314267900284127, 116781203402752052641, -5553985490569476301923

Offset: 0

Author

Seiichi Manyama, Mar 05 2022

Keywords

• sign

Crossrefs

Cf. A352122, A352123.

Cf. A352075.

Programs

Mathematica

m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[-2*x])^(1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(2-exp(-2*x))))

PARI

a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, -2*j+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (-2*j+1)) * Stirling2(n,k).

a(n) ~ (-1)^(n+1) * 2^n * n^(n-1) / (log(2)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Mar 06 2022