A066668 Signed row sums of A066667.

1, 1, 1, -1, -19, -151, -1091, -7841, -56519, -396271, -2442439, -7701409, 145269541, 4833158329, 104056218421, 2002667085119, 37109187217649, 679877731030049, 12440309297451121, 227773259993414719, 4155839606711748061, 74724654677947488521, 1293162252850914402221

Offset: 0

Author

Christian G. Bower, Dec 17 2001

Keywords

• sign

Comments

Numerators in exp(x/(x+1)) power series (signs are different). - Benoit Cloitre, Mar 13 2002

Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i>j, m(i,j)=j-i if j>i. - Vladeta Jovovic, Jan 19 2003

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000262, A111884.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(x-1))/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

Maple

a := n -> n!*hypergeom([1-n], [2], 1):
seq(simplify(a(n)), n=1..19); # Peter Luschny, Mar 30 2015

Mathematica

CoefficientList[Series[E^(x/(x-1))/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2014 *)
Table[Sum[-BellY[n+1, k, -Range[n+1]!], {k, n+1}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x)^2)) \\ G. C. Greubel, May 15 2018

Sage

A066668 = lambda n: (-1)^n*hypergeometric([-n-1,-n-1,-n],[-n-1],-1)
[Integer(A066668(n).n(100)) for n in range(23)] # Peter Luschny, Sep 22 2014

Formula

a(n) = n!LaguerreL(n, 1, 1). - Paul Barry, Sep 08 2004

E.g.f.: exp(x/(x-1))/(1-x)^2.

Conjecture: a(n) +(-2*n+1)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

E.g.f. with a different offset: 1 - product {n >= 1} (1 - x^n)^(phi(n)/n) = x + x^2/2 + x^3/6 - x^4/24 - 19*x^5/120 - ..., where phi(n) = A000010(n) is the Euler totient function. Cf. A000262. - Peter Bala, Jan 01 2014

a(n) = (-1)^n*hypergeom([-n-1,-n-1,-n],[-n-1],-1). - Peter Luschny, Sep 22 2014

a(n) = n!*hypergeom([1-n], [2], 1). - Peter Luschny, Mar 30 2015