A202410 - OEIS

%I #18 Sep 08 2022 08:46:01

%S 1,-2,-1,2,17,94,487,2386,9473,638,-727729,-14280542,-222283631,

%T -3235193378,-46058318473,-649936245646,-9071848025983,

%U -123239922765314,-1562265600970337,-16288001936745662,-55920926830283119,4236297849575724638,201330840708035368199

%N Inverse Lah transform of 1,2,3,...; e.g.f. exp(x/(x-1))*(2*x-1)/(x-1).

%H G. C. Greubel, <a href="/A202410/b202410.txt">Table of n, a(n) for n = 0..449</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations">Sequence transformations</a>.

%F a(n) = Sum_{k=0..n} (-1)^k*(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)* (k+1).

%F a(n) = n!*(L(n,1)-2*L(n-1,1)) for n>0 and a(0)=1. L(n,x) denotes the n-th Laguerre polynomial.

%p A202410_list := proc(n) local k; exp(x/(x-1))*(2*x-1)/(x-1);

%p seq(k!*coeff(series(%,x,n+2),x,k),k=0..n) end: A202410_list(22);

%t Table[If[n==0,1, n! (LaguerreL[n,1] - 2 LaguerreL[n-1,1])], {n,0,20}]

%t With[{nmax = 50}, CoefficientList[Series[Exp[x/(x - 1)]*(2*x - 1)/(x - 1), {x,0,nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, May 23 2018 *)

%o (Sage)

%o def Lah(n, k) :

%o     return (-1)^n*factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)

%o def Lah_invtrans(A) :

%o     L = []

%o     for n in range(len(A)) :

%o         S = sum((-1)^(n-k)*Lah(n,k)*A[k] for k in (0..n))

%o         L.append(S)

%o     return L

%o def A202410_list(n) :

%o     return Lah_invtrans([i for i in (1..n)])

%o A202410_list(20)

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))*(2*x-1)/(x-1))) \\ _G. C. Greubel_, May 23 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(x-1))*(2*x-1)/(x-1))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 23 2018

%Y Cf. A059115.

%K sign

%O 0,2

%A _Peter Luschny_, Jan 18 2012