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A202410
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Inverse Lah transform of 1,2,3,...; e.g.f. exp(x/(x-1))*(2*x-1)/(x-1).
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2
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1, -2, -1, 2, 17, 94, 487, 2386, 9473, 638, -727729, -14280542, -222283631, -3235193378, -46058318473, -649936245646, -9071848025983, -123239922765314, -1562265600970337, -16288001936745662, -55920926830283119, 4236297849575724638, 201330840708035368199
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OFFSET
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0,2
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LINKS
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FORMULA
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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.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
return Lah_invtrans([i for i in (1..n)])
(PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))*(2*x-1)/(x-1))) \\ G. C. Greubel, May 23 2018
(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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