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A182386 a(0) = 1, a(n) = 1 - n * a(n-1).

%I A182386 #56 Sep 08 2022 08:45:55
%S A182386 1,0,1,-2,9,-44,265,-1854,14833,-133496,1334961,-14684570,176214841,
%T A182386 -2290792932,32071101049,-481066515734,7697064251745,-130850092279664,
%U A182386 2355301661033953,-44750731559645106,895014631192902121,-18795307255050944540,413496759611120779881
%N A182386 a(0) = 1, a(n) = 1 - n * a(n-1).
%C A182386 Hankel transform is A055209.
%H A182386 Alois P. Heinz, <a href="/A182386/b182386.txt">Table of n, a(n) for n = 0..450</a>
%H A182386 Shirali Kadyrov, Farukh Mashurov, <a href="https://arxiv.org/abs/1912.03214">Generalized continued fraction expansions for Pi and E</a>, arXiv:1912.03214 [math.NT], 2019.
%F A182386 a(n+2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A001563(k+1) for k = 0, 1, ..., n.
%F A182386 E.g.f.: exp(x) / (1 + x).
%F A182386 a(n) = (-1)^n * A000166(n).
%F A182386 G.f.: 1/U(0) where U(k)=  1 - x + x*(k+1)/(1 + x*(k+1)/U(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Oct 13 2012
%F A182386 E.g.f.: exp(x) / (1 + x) = 1/(1 - x^2/2 + x^3/(U(0) + 2*x))  where U(k)= k^2 + k*(4-x) - 2*x + 3 + x*(k+1)*(k+3)^2/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 16 2012
%F A182386 G.f.: 1/Q(0) where Q(k) =1 + 4*k*x - x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 10 2013
%F A182386 G.f.: 1/Q(0), where Q(k)= 1 + (k+1)^2*(x) - x/(1 - x*(k+1)^2/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 18 2013
%F A182386 a(n) = (-1)^n*Gamma(n+1,-1)*exp(-1), where Gamma(a,x) is the incomplete gamma function. - _Ilya Gutkovskiy_, May 05 2016
%F A182386 0 = a(n)*(-a(n+1) +a(n+2) +a(n+3)) +a(n+1)*(+a(n+1) -2*a(n+2) -a(n+3)) +a(n+2)*(+a(n+2)) for all n>=0. - _Michael Somos_, Jun 26 2018
%e A182386 G.f. = 1 + x^2 - 2*x^3 + 9*x^4 - 44*x^5 + 265*x^6 - 1854*x^7 + 14833*x^8 + ...
%p A182386 a:= proc(n) option remember; `if`(n=0, 1, 1-n*a(n-1)) end:
%p A182386 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 19 2015
%t A182386 a[n_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x]
%t A182386 Table[a[n] /. x -> 1, {n, 0, 20}] (* _Gerry Martens_ , May 05 2016 *)
%t A182386 a[0] = 1; a[n_] := a[n] = 1 - n a[n - 1]; Table[a@ n, {n, 0, 22}] (* _Michael De Vlieger_, May 05 2016 *)
%o A182386 (PARI) {a(n) = if( n<1, n==0, 1 - n * a(n-1))};
%o A182386 (Sage)
%o A182386 A182386 = lambda n: hypergeometric([-n, 1], [], 1)
%o A182386 print([simplify(A182386(n)) for n in range(23)]) # _Peter Luschny_, Oct 19 2014
%o A182386 (Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/(1+x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 11 2018
%Y A182386 Cf. A000166, A001563, A055209.
%K A182386 sign
%O A182386 0,4
%A A182386 _Michael Somos_, Apr 27 2012