A326325 a(n) = 2^n*n!*([z^n] exp(x*z)*tanh(z))(1/2).

0, 2, 4, -10, -56, 362, 2764, -24610, -250736, 2873042, 36581524, -512343610, -7828053416, 129570724922, 2309644635484, -44110959165010, -898621108880096, 19450718635716002, 445777636063460644, -10784052561125704810, -274613643571568682776, 7342627959965776406282

Offset: 0

Author

Peter Luschny, Jun 28 2019

Keywords

• sign

Crossrefs

Cf. A162660, A009832, A155585, A212435.

Programs

Maple

seq(1 - 4^n*euler(n, 1/4), n=0..21);

Mathematica

p := CoefficientList[Series[Exp[x z] Tanh[z], {z, 0, 21}], z];
norm := Table[2^n n!, {n, 0, 21}]; norm  (p /. x -> 1/2)

Formula

a(n) = 1 - 4^n*Euler(n, 1/4).

Let p(n, x) = -x^n + Sum_{k=0..n} binomial(n,k)*Euler(k)*(x+1)^(n-k) (the polynomials defined in A162660), then a(n) = 2^n*p(n, 1/2).