A342161 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..24.

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).

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).

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

PROG

(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.<x> = PowerSeriesRing(QQ, default_prec=prec)

f = (2*exp(2*x)*(exp(x) - 1))/(exp(2*x) + 1)