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A317096
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Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)).
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1
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1, 0, 1, 8, 69, 704, 8485, 118824, 1900297, 34191296, 683657001, 15038537480, 360903291661, 9383240195328, 262727926084429, 7881806223689384, 252217461390469905, 8575390623429206144, 308714050531090308817, 11731134397549023854856, 469245396934886909801941, 19708307298664103361642560
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000166(k)*n!/k!.
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n+1). - Vaclav Kotesovec, Mar 26 2019
a(n) = n!*(LaguarreL(n,1) + Sum_{j=0..n-2} 2^(n-j-2)*LaguerreL(j,1)). - G. C. Greubel, Mar 09 2021
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MAPLE
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a:=series(exp(x/(x - 1))*(1 - x)/(1 - 2*x), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019
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MATHEMATICA
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nmax = 21; CoefficientList[Series[Exp[x/(x - 1)] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Subfactorial[k] n!/k!, {k, 0, n}], {n, 0, 21}]
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PROG
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(Sage)
def A317096(n): return factorial(n)*( gen_laguerre(n, 0, 1) + sum(2^(n-j-2)*gen_laguerre(j, 0, 1) for j in (0..n-2)) )
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 26);
Coefficients(R!(Laplace( ((1-x)/(1-2*x))*Exp(x/(x-1)) ))); // G. C. Greubel, Mar 09 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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