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A006152 Exponential generating function x*exp(x/(1-x)).

%I A006152 M1939 #54 Sep 19 2024 18:07:55
%S A006152 1,2,9,52,365,3006,28357,301064,3549177,45965530,648352001,9888877692,
%T A006152 162112109029,2841669616982,53025262866045,1049180850990736,
%U A006152 21937381717388657,483239096122434354,11184035897992673017,271287473871771163460,6881656485607798743261
%N A006152 Exponential generating function x*exp(x/(1-x)).
%C A006152 a(n) is the number of labeled rooted trees with every non-root vertex of degree 1 or 2. - _Geoffrey Critzer_, May 21 2012.
%C A006152 Total number of unit length lists in all sets of lists, cf. A000262. - _Alois P. Heinz_, May 10 2016
%D A006152 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006152 Vincenzo Librandi, <a href="/A006152/b006152.txt">Table of n, a(n) for n = 1..200</a>
%H A006152 S. Getu and L. W. Shapiro, <a href="/A006152/a006152.pdf">Combinatorial view of the composition of functions</a>, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
%H A006152 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=156">Encyclopedia of Combinatorial Structures 156</a>
%F A006152 a(n) = n*A000262(n-1).
%F A006152 D-finite with recurrence a(n) = 2*(n-1)*a(n-1)-(n^2-5*n+5)*a(n-2)-(n-4)*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 05 2012
%F A006152 a(n) ~ n^(n-1/4)*exp^(2*sqrt(n)-n-1/2)/sqrt(2). - _Vaclav Kotesovec_, Oct 05 2012
%F A006152 a(n) = A320264(n+1,n). - _Alois P. Heinz_, Oct 08 2018
%t A006152 nn = 17; a = x/(1 - x);
%t A006152 Range[0, nn]! CoefficientList[Series[x Exp[a], {x, 0, nn}], x]  (* _Geoffrey Critzer_, May 21 2012 *)
%o A006152 (PARI) a(n)=n!*polcoeff(x*exp(x/(1-x)+O(x^n)), n)
%Y A006152 Cf. A000262, A320264.
%K A006152 nonn,easy
%O A006152 1,2
%A A006152 _Simon Plouffe_
%E A006152 More terms from _Michael Somos_, Jun 07 2000