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a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.
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%I #106 Sep 06 2023 01:17:08

%S 0,1,2,8,40,224,1344,8448,54912,366080,2489344,17199104,120393728,

%T 852017152,6085836800,43818024960,317680680960,2317200261120,

%U 16992801914880,125210119372800,926554883358720,6882979133521920

%N a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

%C A151374 shifted one place right. - _Joerg Arndt_, Mar 17 2011

%C The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - _Valery A. Liskovets_, Mar 17 2005

%C This is also the number of strings of length 2n-2 of two different types of balanced parentheses. For example, a(2) = 2, since the two possible strings of length 2 are [] and (), a(3) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][]. - _Jeffrey Shallit_, Jun 03 2006

%C Row sums of number triangle A110506. - _Paul Barry_, Jul 24 2005

%C Also row sums of triangle in A085880. - _Philippe Deléham_, Aug 01 2005

%C Row sums of number triangle A114608. - _Philippe Deléham_, Oct 15 2008

%D V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

%H G. C. Greubel, <a href="/A052701/b052701.txt">Table of n, a(n) for n = 0..1000</a>

%H Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, <a href="https://arxiv.org/abs/2309.00426">A lattice on Dyck paths close to the Tamari lattice</a>, arXiv:2309.00426 [math.CO], 2023.

%H M. Bousquet-Mélou, <a href="https://arxiv.org/abs/math/0501266">Limit laws for embedded trees</a>, arXiv:math/0501266 [math.CO], 2005.

%H Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont, and Światosław R. Gal, <a href="https://arxiv.org/abs/2104.14530">Reflection length with two parameters in the asymptotic representation theory of type B/C and applications</a>, arXiv:2104.14530 [math.RT], 2021.

%H F. Chapoton and S. Giraudo, <a href="http://arxiv.org/abs/1310.4521">Enveloping operads and bicoloured noncrossing configurations</a>, arXiv:1310.4521 [math.CO], 2013.

%H Ivan Geffner and Marc Noy, <a href="https://doi.org/10.37236/6249">Counting Outerplanar Maps</a>, Electronic Journal of Combinatorics, Vol. 24, No. 2 (2017), Article P2.3.

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=651">Encyclopedia of Combinatorial Structures 651</a>.

%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., Vol. 36, No.4 (2006), pp. 364-387.

%H Vincent Pilaud and V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.

%H <a href="/index/Res#revert">Index to sequences related to reversion of series</a>.

%F a(n) = A052714(n)/n!.

%F a(n) = A003645(n-2)*2, n>1.

%F a(n) = 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.

%F D-finite with recurrence: a(1)=1, (-4+8*n)*a(n) - (n+1)*a(n+1) = 0.

%F G.f.: (1-sqrt(1-8*x))/4 = x*C(2*x) where C(x) is g.f. for Catalan numbers, A000108.

%F G.f. A(x) satisfies 2*A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2*A(x)^2=x/(1-2*A(x)). Series reversion of x-2*x^2. - _Michael Somos_, Sep 06 2003

%F a(0)=0, a(1)=1; a(n) = 2*Sum_{i=1..n-1} a(i)*a(n-i). - _Benoit Cloitre_, Mar 16 2004

%F With a different offset, a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} (j*C(2n-j-1, n-j)*C(j, k)*2^(n-j)/n), n>0. - _Paul Barry_, Jul 24 2005

%F The Hankel transform of a(n+1) = [1,2,8,40,224,1344,...] is 4^C(n+1,2). - _Philippe Deléham_, Nov 06 2007

%F G.f.: x + 4*x^2/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1) ; (continued fraction ). - _Sergei N. Gladkovskii_, Apr 05 2013

%F a(n) ~ 8^(n-1)/(sqrt(Pi)*n^(3/2)). - _Ilya Gutkovskiy_, Dec 04 2016

%F From _Amiram Eldar_, Mar 06 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 68/49 + 96*arcsin(sqrt(1/8))/(49*sqrt(7)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 20/27 - 16*log(2)/81. (End)

%p spec := [S,{B=Union(C,Z),S=Union(B,C),C=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t InverseSeries[Series[y-2*y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* _Len Smiley_, Apr 07 2000 *)

%t Join[{0},Table[2^n CatalanNumber[n],{n,0,30}]] (* _Harvey P. Dale_, Aug 29 2015 *)

%o (PARI) a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!/n!)

%o (PARI) a(n)=if(n<1,0,polcoeff(serreverse(x-2*x^2+x*O(x^n)),n))

%o (PARI) a(n)=if(n<1,0,polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))),n))

%Y Limit of array A102544.

%Y Cf. A000108, A003645, A052714, A195699.

%K easy,nonn

%O 0,3

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

%E Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001

%E Additional comments from _Michael Somos_, Feb 24 2002