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A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

%I A052852 #102 Jun 02 2025 16:49:53
%S A052852 0,1,4,21,136,1045,9276,93289,1047376,12975561,175721140,2581284541,
%T A052852 40864292184,693347907421,12548540320876,241253367679185,
%U A052852 4909234733857696,105394372192969489,2380337795595885156,56410454014314490981,1399496554158060983080
%N A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).
%C A052852 A simple grammar.
%C A052852 Number of {121,212}-avoiding n-ary words of length n. - _Ralf Stephan_, Apr 20 2004
%C A052852 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
%C A052852 a(n) is the total number of components summed over all nilpotent partial permutations of [n]. - _Geoffrey Critzer_, Feb 19 2022
%H A052852 Vincenzo Librandi, <a href="/A052852/b052852.txt">Table of n, a(n) for n = 0..200</a>
%H A052852 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, Section 2.
%H A052852 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=820">Encyclopedia of Combinatorial Structures 820</a>
%H A052852 Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.
%H A052852 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 A052852 Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%H A052852 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 A052852 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A052852 D-finite with recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.
%F A052852 a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - _Wolfdieter Lang_, Jun 19 2001
%F A052852 a(n) = n*A002720(n-1). [Riordan] - _Vladeta Jovovic_, Mar 18 2005
%F A052852 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 A052852 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 A052852 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 A052852 a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 09 2012
%F A052852 a(n) = n!*hypergeom([-n+1], [1], -1) for n>=1. - _Peter Luschny_, Oct 18 2014 [Simplified by _Natalia L. Skirrow_, 30 December 2024]
%F A052852 a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - _Peter Luschny_, Oct 18 2014
%F A052852 a(n) = n!*LaguerreL(n-1, 0, -1) for n>=1. - _Peter Luschny_, Apr 08 2015, simplified Dec 30 2024
%F A052852 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
%F A052852 Equals column A059114(n, 2) for n >= 1. - _G. C. Greubel_, Feb 23 2021
%F A052852 a(n) = Sum_{k=1..n} k * A271703(n,k). - _Geoffrey Critzer_, Feb 19 2022
%p A052852 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 A052852 a := n -> ifelse(n = 0, 0, n!*hypergeom([-n+1], [1], -1)): seq(simplify(a(n)), n = 0..18);  # _Peter Luschny_, Dec 30 2024
%t A052852 Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 09 2012 *)
%t A052852 Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* _G. C. Greubel_, Feb 23 2021 *)
%o A052852 (PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ _G. C. Greubel_, May 15 2018
%o A052852 (Sage) [0 if n==0 else factorial(n)*gen_laguerre(n-1, 0, -1) for n in (0..30)] # _G. C. Greubel_, Feb 23 2021
%o A052852 (Magma) [n eq 0 select 0 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), -1): n in [0..30]]; // _G. C. Greubel_, Feb 23 2021
%Y A052852 Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).
%Y A052852 Cf. A000262. A000169, A000312.
%Y A052852 Cf. A059114, A288268.
%K A052852 easy,nonn
%O A052852 0,3
%A A052852 encyclopedia(AT)pommard.inria.fr, Jan 25 2000