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A201720 The total number of components in (A011800) of all labeled forests on n nodes whose components are all paths.

%I A201720 #16 Mar 28 2023 09:54:24
%S A201720 0,1,3,12,64,420,3246,28798,288072,3205044,39234340,523821936,
%T A201720 7572221328,117792884872,1961516974704,34807390821960,655594811020096,
%U A201720 13060711726818768,274358217793164912,6060159633360214144,140404595387426964480
%N A201720 The total number of components in (A011800) of all labeled forests on n nodes whose components are all paths.
%F A201720 E.g.f.: x*(2-x)*exp[x*(2-x)/(2-2x)]/(2-2x). - _R. J. Mathar_, Jun 27 2022
%F A201720 D-finite with recurrence 6*(n+1)*a(n) +2*(-6*n^2-19*n+35)*a(n-1) +2*(3*n^3+26*n^2-102*n+75)*a(n-2) -(n-2)*(29*n^2-102*n+85)*a(n-3) +(13*n-15)*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jun 27 2022
%p A201720 A201720 := proc(n)
%p A201720     g := (2*x-x^2)*exp((2*x-x^2)/(2-2*x))/(2-2*x) ;
%p A201720     coeftayl(g,x=0,n) ;
%p A201720     %*n! ;
%p A201720 end proc:
%p A201720 seq(A201720(n),n=0..30) ; # _R. J. Mathar_, Jun 27 2022
%t A201720 D[Range[0, 20]! CoefficientList[ Series[Exp[y (2 x - x^2)/(2 - 2 x)], {x, 0, 20}], x], y] /. y -> 1
%Y A201720 Cf. A011800.
%K A201720 nonn
%O A201720 0,3
%A A201720 _Geoffrey Critzer_, Dec 04 2011