A342161 Expansion of the exponential generating function (tanh(x) - sech(x) + 1) * exp(x).

0, 1, 3, 4, -3, -14, 63, 274, -1383, -7934, 50523, 353794, -2702763, -22368254, 199360983, 1903757314, -19391512143, -209865342974, 2404879675443, 29088885112834, -370371188237523, -4951498053124094, 69348874393137903, 1015423886506852354, -15514534163557086903

Offset: 0

Author

Petros Hadjicostas, Mar 03 2021

Keywords

• sign

Links

• A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (zeroth column of matrix b_{n,k} on p. 19).

Crossrefs

Cf. A000111, A323834.

Programs

Maple

series((2*exp(x)-2)/(exp(-2*x)+1),x,30):seq(n!*coeff(%,x,n),n=0..24); # Peter Luschny, Mar 05 2021

PARI

my(x='x+O('x^30)); concat(0, Vec(serlaplace((-1/cosh(x) + tanh(x) + 1)*exp(x)))) \\ Michel Marcus, Mar 05 2021

SageMath

def A323834List(prec):
    R. = PowerSeriesRing(QQ, default_prec=prec)
    f = (2*exp(2*x)*(exp(x) - 1))/(exp(2*x) + 1)
    return f.egf_to_ogf().list()
print(A323834List(25)) # Peter Luschny, Mar 05 2021

Formula

a(n) = A323834(n, 0).

a(n) = n! [x^n] (tanh(x) - sech(x) + 1) * exp(x).

a(n) = Sum_{i=1..n} binomial(n,i) * (-1)^floor((i-1)/2) * A000111(i).