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 A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)). 33
 0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A simple grammar. Number of {121,212}-avoiding n-ary words of length n. - Ralf Stephan, Apr 20 2004 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911, Section 2. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 820 Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006. John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers. Luis Verde-Star A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1. Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13. FORMULA 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. a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - Wolfdieter Lang, Jun 19 2001 a(n) = n*A002720(n-1). [Riordan] - Vladeta Jovovic, Mar 18 2005 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 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 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 a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012 a(n) = (n+1)!*hypergeom([-n+1], , -1)/2 for n>=1. - Peter Luschny, Oct 18 2014 a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014 a(n) = (n-1)!*LaguerreL(n-1, 2, -1) for n>=1. - Peter Luschny, Apr 08 2015 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 Equals column A059114(n, 2) for n >= 1. - G. C. Greubel, Feb 23 2021 MAPLE spec := [S, {B=Set(C), C=Sequence(Z, 1 <= card), S=Prod(B, C)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); a := n -> `if`(n=0, 0, (n+1)!*hypergeom([-n+1], , -1)/2); seq(simplify(a(n)), n=0..18); # Peter Luschny, Oct 18 2014 MATHEMATICA Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 09 2012 *) Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *) PROG (PARI) my(x='x+O('x^30)); concat(, Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ G. C. Greubel, May 15 2018 (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 (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 CROSSREFS Row sums of unsigned triangle A062139 (generalized a=2 Laguerre). Cf. A000262. A000169, A000312. Cf. A059114. Sequence in context: A205077 A292928 A209881 * A288869 A288268 A265952 Adjacent sequences:  A052849 A052850 A052851 * A052853 A052854 A052855 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 STATUS approved

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Last modified September 23 14:40 EDT 2021. Contains 347618 sequences. (Running on oeis4.)