A119884 Expansion of e.g.f. sech(x)/(1-x).

1, 1, 1, 3, 17, 85, 449, 3143, 26529, 238761, 2337089, 25707979, 311198513, 4045580669, 56438768385, 846581525775, 13564695924545, 230599830717265, 4148392073235329, 78819449391471251, 1576759359017662545, 33111946539370913445, 728393474991766957889

Offset: 0

Author

Paul Barry, May 26 2006

Keywords

• easy
• nonn

Comments

Transform of n! under the matrix A119879.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A119879, A122045.

Programs

Magma

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

Maple

restart: G(x):= sech(x)/(1-x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

Mathematica

CoefficientList[Series[1/((1-x)*(E^x/2+E^(-x)/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)
Table[n!*Sum[EulerE[j]/j!, {j,0,n}], {n,0,40}] (* G. C. Greubel, Jun 07 2023 *)

SageMath

def A119884(n): return factorial(n)*sum(euler_number(j)/factorial(j) for j in range(n+1))
[A119884(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

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

E.g.f.: 1/U(0) where U(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2012

a(n) ~ n! * 2*exp(1)/(exp(2)+1). - Vaclav Kotesovec, Sep 25 2013

a(n) = n! * Sum_{j=0..n} A122045(j)/j!. - G. C. Greubel, Jun 07 2023