A029767 - OEIS

%I #55 Sep 08 2022 08:44:50

%S 0,1,3,14,90,744,7560,91440,1285200,20603520,371226240,7428153600,

%T 163459296000,3923502105600,102017281766400,2856571067750400,

%U 85698439706880000,2742370993410048000,93240969463369728000,3356681303055015936000,127554011161191014400000

%N a(n) = (n-1)!*(2^n-1) for n>=1, a(0)=0.

%C Labeled octupi with n nodes.

%C a(n) is the number of connected endofunctions on n points such that every nonrecurrent element has at most one element in its preimage and every recurrent element has at most two elements in its preimage. - _Geoffrey Critzer_, Dec 07 2011

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, pp. 12, 55, 409.

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.1.5.

%H G. C. Greubel, <a href="/A029767/b029767.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=498">Encyclopedia of Combinatorial Structures 498</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=777">Encyclopedia of Combinatorial Structures 777</a>

%F E.g.f.: log(1-x)-log(1-2*x).

%F In Maple notation, representation as an infinite sum: a(n) = sum((n+k)!/((k+1)! *2^k), k=0..infinity)/2, n=1, 2... Representation as n-th moment of a positive function on a positive half-axis: a(n)=int(x^n*1/2*exp(-x)/x*(2*exp(1/2*x)-2), x=0..infinity), n=1, 2... - _Karol A. Penson_, Oct 15 2002

%F D-finite with recurrence: a(n) +3*(-n+1)*a(n-1) +2*(n-1)*(n-2)*a(n-2) = 0. - _R. J. Mathar_, Jan 08 2013

%F a(n) = n!*Sum_{k=0..n-1} binomial(n-1,k)/(k+1). - _J. M. Bergot_, Jul 30 2015

%F a(n) = (1/zeta(n)) * Integral_{x=0..1} (log(1/x))^(n-1) / (sqrt(x) * (1-x)) dx. - _Amrik Singh Nimbran_, May 06 2018

%p with(combinat): seq(stirling1(j,1)*stirling2(j+1,2)*(-1)^(j+1), j=0..16); # _Zerinvary Lajos_, Mar 30 2007

%t a=x/(1-x); Range[0,20]! CoefficientList[Series[Log[1/(1-a)], {x,0,20}], x]  (* _Geoffrey Critzer_, Dec 07 2011 *)

%t Join[{0}, Table[(n - 1)! (2^n - 1), {n, 20}]] (* _Vincenzo Librandi_, Apr 18 2015 *)

%o (Magma) [0] cat [Factorial(n-1)*(2^n-1): n in [1..20]]; // _Vincenzo Librandi_, Apr 18 2015

%o (PARI) concat([0], for(n=1,25, print1((n-1)!*(2^n -1), ", "))) \\ _G. C. Greubel_, Jan 19 2017

%o (GAP) Concatenation([0],List([1..20],n->Factorial(n-1)*(2^n-1))); # _Muniru A Asiru_, Aug 09 2018

%Y Cf. A001865.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_