A119882 Expansion of e.g.f.: (1+x)*sech(x).

1, 1, -1, -3, 5, 25, -61, -427, 1385, 12465, -50521, -555731, 2702765, 35135945, -199360981, -2990414715, 19391512145, 329655706465, -2404879675441, -45692713833379, 370371188237525, 7777794952988025, -69348874393137901, -1595024111042171723, 15514534163557086905

Offset: 0

Author

Paul Barry, May 26 2006

Keywords

• easy
• sign

Comments

Transform of binomial(1,n) under the matrix A119879.

Links

• Robert Israel, Table of n, a(n) for n = 0..480

Crossrefs

Cf. A119879, A122045.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1+x)/Cosh(x) ))); // G. C. Greubel, Jun 07 2023

Maple

seq(`if`(n::odd, n*euler(n-1), euler(n)), n=0..24); # Peter Luschny, May 30 2016

Mathematica

Table[EulerE[n] + n*EulerE[n-1], {n,20}] (* Benedict W. J. Irwin, May 30 2016 *)
With[{nn=30},CoefficientList[Series[(1+x)Sech[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 22 2025 *)

PARI

Vec(serlaplace((1+x)/cosh(x + O(x^30)))) \\ Andrew Howroyd, Feb 27 2018

SageMath

def A119882(n): return n*euler_number(n-1) if n%2==1 else euler_number(n)
[A119882(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

a(n) = Sum_{k=0..n} A119879(n,k)*C(1,k).

E.g.f.: (1+x)/sech(x) = (1+x)*(1 - x^2/Q(0)), where Q(k) = (2*k+1)*(2*k+2) + x^2 - (2*k+1)*(2*k+2)*x^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013

a(n) = EulerE[n] + n*EulerE[n-1], n>0. - Benedict W. J. Irwin, May 30 2016