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%I #46 Sep 08 2022 08:45:04
%S 1,1,1,-1,-19,-151,-1091,-7841,-56519,-396271,-2442439,-7701409,
%T 145269541,4833158329,104056218421,2002667085119,37109187217649,
%U 679877731030049,12440309297451121,227773259993414719,4155839606711748061,74724654677947488521,1293162252850914402221
%N Signed row sums of A066667.
%C Numerators in exp(x/(x+1)) power series (signs are different). - _Benoit Cloitre_, Mar 13 2002
%C 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
%H Vincenzo Librandi, <a href="/A066668/b066668.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F a(n) = n!LaguerreL(n, 1, 1). - _Paul Barry_, Sep 08 2004
%F E.g.f.: exp(x/(x-1))/(1-x)^2.
%F Conjecture: a(n) +(-2*n+1)*a(n-1) +n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 26 2012
%F 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
%F a(n) = (-1)^n*hypergeom([-n-1,-n-1,-n],[-n-1],-1). - _Peter Luschny_, Sep 22 2014
%F a(n) = n!*hypergeom([1-n], [2], 1). - _Peter Luschny_, Mar 30 2015
%p a := n -> n!*hypergeom([1-n], [2], 1):
%p seq(simplify(a(n)), n=1..19); # _Peter Luschny_, Mar 30 2015
%t CoefficientList[Series[E^(x/(x-1))/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 13 2014 *)
%t Table[Sum[-BellY[n+1, k, -Range[n+1]!], {k, n+1}], {n, 0, 25}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o (Sage)
%o A066668 = lambda n: (-1)^n*hypergeometric([-n-1,-n-1,-n],[-n-1],-1)
%o [Integer(A066668(n).n(100)) for n in range(23)] # _Peter Luschny_, Sep 22 2014
%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x)^2)) \\ _G. C. Greubel_, May 15 2018
%o (Magma) m:=25; R<x>:=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
%Y Cf. A000262, A111884.
%K sign
%O 0,5
%A _Christian G. Bower_, Dec 17 2001
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