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A052852 E.g.f.: (x/(1-x))*exp(x/(1-x)). 32

%I

%S 0,1,4,21,136,1045,9276,93289,1047376,12975561,175721140,2581284541,

%T 40864292184,693347907421,12548540320876,241253367679185,

%U 4909234733857696,105394372192969489,2380337795595885156

%N E.g.f.: (x/(1-x))*exp(x/(1-x)).

%C A simple grammar.

%C Number of {121,212}-avoiding n-ary words of length n. - _Ralf Stephan_, Apr 20 2004

%C The infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n) when n is a positive integer. - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008

%D D. Angell, A family of continued fractions, J. Numb. Theory 130 (2010) 904-911 doi:10.1016/j.jnt.2009.12.003, Section 2.

%H Vincenzo Librandi, <a href="/A052852/b052852.txt">Table of n, a(n) for n = 0..200</a>

%H David Angell, <a href="http://dx.doi.org/10.1016/j.jnt.2009.12.003">A family of continued fractions</a>, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=820">Encyclopedia of Combinatorial Structures 820</a>

%H F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.

%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.

%H Michael Wallner, <a href="https://arxiv.org/abs/1706.07163">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F Recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.

%F a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - _Wolfdieter Lang_, Jun 19 2001

%F a(n) = n*A002720(n-1). [Riordan] - _Vladeta Jovovic_, Mar 18 2005

%F Related to an n-dimensional series : for n>=1, a(n)=(n!/e)* sum_{k_n>=k_{n-1}>=...>=k_1>=0}1/(k_n)!). - _Benoit Cloitre_, Sep 30 2006

%F E.g.f.: (x/(1-x))*exp((x/(1-x))) =(x/(1-x))*G(0); G(k)=1+x/((2*k+1)*(1-x)-x*(1-x)*(2*k+1)/(x+(1-x)*(2*k+2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 24 2011

%F a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000262 and A005493. - _Peter Bala_, Nov 25 2011

%F a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 09 2012

%F a(n) = (n+1)!*hypergeom([-n+1], [3], -1)/2 for n>=1. - _Peter Luschny_, Oct 18 2014

%F a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - _Peter Luschny_, Oct 18 2014

%F a(n) = (n-1)!*LaguerreL(n-1, 2, -1) for n>=1. - _Peter Luschny_, Apr 08 2015

%F The series reversion of the e.g.f. equals W(x)/(1 + W(x)) = x - 2^2*x^2/2! + 3^3*x^3/3! - 4^4*x^4/4! + ..., essentially the e.g.f. for a signed version of A000312, where W(x) is Lambert's W-function (see A000169). - _Peter Bala_, Jun 14 2016

%p spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p a := n -> `if`(n=0, 0, (n+1)!*hypergeom([-n+1], [3], -1)/2); seq(simplify(a(n)), n=0..18); # _Peter Luschny_, Oct 18 2014

%t Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 09 2012 *)

%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ _G. C. Greubel_, May 15 2018

%Y Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).

%Y Cf. A000262. A000169, A000312.

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)