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A052701
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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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18
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0, 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920
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OFFSET
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0,3
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COMMENTS
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The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
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
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REFERENCES
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V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
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LINKS
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Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics, Vol. 24, No. 2 (2017), Article P2.3.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
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FORMULA
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a(n) = 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.
D-finite with recurrence: a(1)=1, (-4+8*n)*a(n) - (n+1)*a(n+1) = 0.
G.f.: (1-sqrt(1-8*x))/4 = x*C(2*x) where C(x) is g.f. for Catalan numbers, A000108.
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
a(0)=0, a(1)=1; a(n) = 2*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
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
The Hankel transform of a(n+1) = [1,2,8,40,224,1344,...] is 4^C(n+1,2). - Philippe Deléham, Nov 06 2007
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
Sum_{n>=1} 1/a(n) = 68/49 + 96*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20/27 - 16*log(2)/81. (End)
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MAPLE
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spec := [S, {B=Union(C, Z), S=Union(B, C), C=Prod(S, S)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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InverseSeries[Series[y-2*y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)
Join[{0}, Table[2^n CatalanNumber[n], {n, 0, 30}]] (* Harvey P. Dale, Aug 29 2015 *)
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PROG
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(PARI) a(n)=if(n<1, 0, 2^(n-1)*(2*n-2)!/(n-1)!/n!)
(PARI) a(n)=if(n<1, 0, polcoeff(serreverse(x-2*x^2+x*O(x^n)), n))
(PARI) a(n)=if(n<1, 0, polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))), n))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001
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STATUS
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approved
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