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

1, -2, -1, 2, 17, 94, 487, 2386, 9473, 638, -727729, -14280542, -222283631, -3235193378, -46058318473, -649936245646, -9071848025983, -123239922765314, -1562265600970337, -16288001936745662, -55920926830283119, 4236297849575724638, 201330840708035368199

Offset: 0

Author

Peter Luschny, Jan 18 2012

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449
• Peter Luschny, Sequence transformations.

Crossrefs

Cf. A059115.

Programs

Magma

m:=25; R:=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

Maple

A202410_list := proc(n) local k; exp(x/(x-1))*(2*x-1)/(x-1);
seq(k!*coeff(series(%,x,n+2),x,k),k=0..n) end: A202410_list(22);

Mathematica

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

PARI

x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))*(2*x-1)/(x-1))) \\ G. C. Greubel, May 23 2018

Sage

def Lah(n, k) :
    return (-1)^n*factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
def Lah_invtrans(A) :
    L = []
    for n in range(len(A)) :
        S = sum((-1)^(n-k)*Lah(n,k)*A[k] for k in (0..n))
        L.append(S)
    return L
def A202410_list(n) :
    return Lah_invtrans([i for i in (1..n)])
A202410_list(20)

Formula

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

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.