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A202410 Inverse Lah transform of 1,2,3,...; e.g.f. exp(x/(x-1))*(2*x-1)/(x-1).

%I A202410 #18 Sep 08 2022 08:46:01
%S A202410 1,-2,-1,2,17,94,487,2386,9473,638,-727729,-14280542,-222283631,
%T A202410 -3235193378,-46058318473,-649936245646,-9071848025983,
%U A202410 -123239922765314,-1562265600970337,-16288001936745662,-55920926830283119,4236297849575724638,201330840708035368199
%N A202410 Inverse Lah transform of 1,2,3,...; e.g.f. exp(x/(x-1))*(2*x-1)/(x-1).
%H A202410 G. C. Greubel, <a href="/A202410/b202410.txt">Table of n, a(n) for n = 0..449</a>
%H A202410 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations">Sequence transformations</a>.
%F A202410 a(n) = Sum_{k=0..n} (-1)^k*(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)* (k+1).
%F A202410 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 A202410_list := proc(n) local k; exp(x/(x-1))*(2*x-1)/(x-1);
%p A202410 seq(k!*coeff(series(%,x,n+2),x,k),k=0..n) end: A202410_list(22);
%t A202410 Table[If[n==0,1, n! (LaguerreL[n,1] - 2 LaguerreL[n-1,1])], {n,0,20}]
%t A202410 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 A202410 (Sage)
%o A202410 def Lah(n, k) :
%o A202410     return (-1)^n*factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
%o A202410 def Lah_invtrans(A) :
%o A202410     L = []
%o A202410     for n in range(len(A)) :
%o A202410         S = sum((-1)^(n-k)*Lah(n,k)*A[k] for k in (0..n))
%o A202410         L.append(S)
%o A202410     return L
%o A202410 def A202410_list(n) :
%o A202410     return Lah_invtrans([i for i in (1..n)])
%o A202410 A202410_list(20)
%o A202410 (PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))*(2*x-1)/(x-1))) \\ _G. C. Greubel_, May 23 2018
%o A202410 (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 A202410 Cf. A059115.
%K A202410 sign
%O A202410 0,2
%A A202410 _Peter Luschny_, Jan 18 2012