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%I #27 Nov 17 2020 14:12:46
%S 1,2,5,19,97,621,4771,42673,434673,4959433,62569891,863989941,
%T 12949163833,209203422013,3622195815603,66881477554921,
%U 1311357008557921,27202303879342353,595035842021131843,13686018793997390173,330130829894262294441,8332243981937569166341
%N Expansion of e.g.f. x/(1 - x) + exp(x/(1 - x)).
%C Previous name was: A simple grammar.
%H Alois P. Heinz, <a href="/A052866/b052866.txt">Table of n, a(n) for n = 0..444</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=835">Encyclopedia of Combinatorial Structures 835</a>
%F E.g.f.: (-x-exp(-x/(-1+x))+exp(-x/(-1+x))*x)/(-1+x).
%F Recurrence: {a(0)=1, a(1)=2, a(2)=5, (-n^3-2*n-3*n^2)*a(n)+(3*n^2+10*n+8)*a(n+1)+(-3*n-7)*a(n+2)+a(n+3)}.
%F a(n) = n! + A000262(n) for n>0. - _Vladeta Jovovic_, Nov 29 2002
%p spec := [S,{B=Sequence(Z,1 <= card),C=Set(B),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n=0, 1, add(
%p b(n-j)*binomial(n-1, j-1)*j!, j=1..n))
%p end:
%p a:= n-> `if`(n=0, 1, b(n)+n!):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 10 2016
%t a[n_] := If[n==0, 1, n! (1+Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}])];
%t a /@ Range[0, 21] (* _Jean-François Alcover_, Nov 17 2020 *)
%t Flatten[{1, Table[n!*(1 + Hypergeometric1F1[1 - n, 2, -1]), {n, 1, 21}]}] (* _Vaclav Kotesovec_, Nov 17 2020 *)
%t nmax = 20; CoefficientList[Series[x/(1 - x) + E^(x/(1 - x)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 17 2020 *)
%Y Cf. A000142, A000262.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name, using e.g.f., from _Michel Marcus_, Nov 17 2020
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