A119883 Expansion of E.g.f. (1 + 2*x + x^2/2) * sech(x).

1, 2, 0, -6, -1, 50, 14, -854, -323, 24930, 11804, -1111462, -631621, 70271890, 46590634, -5980829430, -4531805575, 659311412930, 562021682744, -91385427666758, -86555950096265, 15555589905976050, 16206870089730374, -3190048222084343446, -3625755168948973771

Offset: 0

Author

Paul Barry, May 26 2006

Keywords

• easy
• sign

Comments

Transform of binomial(2,n) under the matrix A119879.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..480

Crossrefs

Cf. A119879, A122045.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1+2*x+x^2/2) /Cosh(x) ))); // G. C. Greubel, Jun 07 2023

Mathematica

Table[If[n<2, n+1, EulerE[n] +2*n*EulerE[n-1] +n*(n-1)*EulerE[n-2]/2], {n,0,30}] (* Benedict W. J. Irwin, May 30 2016 *)
With[{nn=30},CoefficientList[Series[(1+2x+x^2/2)Sech[x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 01 2018 *)

PARI

my(x='x+O('x^44)); Vec(serlaplace((1 + 2*x + x^2/2) / cosh(x))) \\ Joerg Arndt, Jun 01 2016

SageMath

E=euler_number
def A119883(n):
    if n<2: return n+1
    else: return E(n) +2*n*E(n-1) +binomial(n,2)*E(n-2)
[A119883(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

E.g.f.: (1 + 2*x + x^2/2) * sech(x).

a(n) = Sum_{k=0..n} A119879(n,k)*C(2,k).

a(n) = EulerE(n) + 2*n*EulerE(n-1) + n*(n-1)*EulerE(n-2)/2, n>1. - Benedict W. J. Irwin, May 30 2016