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A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

%I A317365 #13 Mar 06 2021 03:32:17
%S A317365 0,1,0,-3,16,-75,336,-1295,1632,55881,-1124000,16722981,-229985040,
%T A317365 3089923837,-41225160144,545880027225,-7069180940864,86130735547665,
%U A317365 -882387869940288,3847692639294541,171852333163131600,-8392137456287472699,276055495385982856720,-8067943451470397940543
%N A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).
%C A317365 Inverse Lah transform of the nonnegative integers (A001477).
%H A317365 G. C. Greubel, <a href="/A317365/b317365.txt">Table of n, a(n) for n = 0..440</a>
%H A317365 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A317365 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A317365 a(n) = Sum_{k=1..n} binomial(n-1,k-1)*n!/(k-1)!.
%F A317365 From _G. C. Greubel_, Mar 05 2021: (Start)
%F A317365 a(n) = n! * Hypergeometric1F1([-(n-1)], [1], -1).
%F A317365 a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1). (End)
%p A317365 a:= proc(n) option remember; add((-1)^(n-k)*
%p A317365       n!/(k-1)!*binomial(n-1, k-1), k=1..n)
%p A317365     end:
%p A317365 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 26 2018
%t A317365 nmax = 23; CoefficientList[Series[x Exp[x/(1 + x)]/(1 + x) , {x, 0, nmax}], x] Range[0, nmax]!
%t A317365 Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] n!/(k - 1)!, {k, n}], {n, 0, 23}]
%t A317365 Join[{0}, Table[(-1)^(n+1) n! LaguerreL[n-1, 1], {n, 23}]]
%o A317365 (Sage) [0 if n==0 else (-1)^(n+1)*factorial(n)*gen_laguerre(n-1, 0, 1) for n in (0..25)] # _G. C. Greubel_, Mar 05 2021
%o A317365 (Magma) [n eq 0 select 0 else (-1)^(n+1)*Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), 1): n in [0..25]]; // _G. C. Greubel_, Mar 05 2021
%o A317365 (PARI) a(n) = (-1)^(n+1)*n!*pollaguerre(n-1,0,1); \\ _Michel Marcus_, Mar 06 2021
%Y A317365 Cf. A001477, A009940, A052852.
%K A317365 sign
%O A317365 0,4
%A A317365 _Ilya Gutkovskiy_, Jul 26 2018