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Exponential generating function x*exp(x/(1-x)).
(Formerly M1939)
8

%I M1939 #54 Sep 19 2024 18:07:55

%S 1,2,9,52,365,3006,28357,301064,3549177,45965530,648352001,9888877692,

%T 162112109029,2841669616982,53025262866045,1049180850990736,

%U 21937381717388657,483239096122434354,11184035897992673017,271287473871771163460,6881656485607798743261

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

%C 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 Total number of unit length lists in all sets of lists, cf. A000262. - _Alois P. Heinz_, May 10 2016

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006152/b006152.txt">Table of n, a(n) for n = 1..200</a>

%H 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=156">Encyclopedia of Combinatorial Structures 156</a>

%F a(n) = n*A000262(n-1).

%F 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 a(n) ~ n^(n-1/4)*exp^(2*sqrt(n)-n-1/2)/sqrt(2). - _Vaclav Kotesovec_, Oct 05 2012

%F a(n) = A320264(n+1,n). - _Alois P. Heinz_, Oct 08 2018

%t nn = 17; a = x/(1 - x);

%t Range[0, nn]! CoefficientList[Series[x Exp[a], {x, 0, nn}], x] (* _Geoffrey Critzer_, May 21 2012 *)

%o (PARI) a(n)=n!*polcoeff(x*exp(x/(1-x)+O(x^n)), n)

%Y Cf. A000262, A320264.

%K nonn,easy

%O 1,2

%A _Simon Plouffe_

%E More terms from _Michael Somos_, Jun 07 2000