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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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14
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Apparently A151374 shifted one place right [Joerg Arndt, Mar 17 2011].
The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), 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 (shallit(AT)graceland.uwaterloo.ca), Jun 03 2006
Row sums of number triangle A110506. - Paul Barry (pbarry(AT)wit.ie), Jul 24 2005
Also row sums of triangle in A085880 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 01 2005
Row sums of number triangle A114608. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 15 2008]
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REFERENCES
| Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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
| M. Bousquet-Melou, Limit laws for embedded trees
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 651
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.
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FORMULA
| 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.
Recurrence: {a(1)=1, (-4+8*n)*a(n)-(n+1)*a(n+1)}
G.f.: (1-sqrt(1-8x))/4 = xC(2x) where C(x) is g.f. for Catalan numbers, A000108.
G.f. A(x) satisfies 2A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2A(x)^2=x/(1-2A(x)). - 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 (benoit7848c(AT)orange.fr), 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 (pbarry(AT)wit.ie), Jul 24 2005
With a different offset, a(n) = 2^n*A000108(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 31 2005
The Hankel transform of a(n+1)=[1,2,8,40,224,1344,...] is 4^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 06 2007
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MAPLE
| 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
| InverseSeries[Series[y-2*y^2], {y, 0, 24}], x] (* then A(x)=y(x) *) - Len Smiley Apr 07 2000
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PROG
| (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
| a(n)=A052714(n)/n!. a(n)=A003645(n-2)*2, n>1.
Limit of array A102544.
Cf. A003645.
Sequence in context: A006195 A092807 A074601 * A151374 A177408 A085485
Adjacent sequences: A052698 A052699 A052700 * A052702 A052703 A052704
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KEYWORD
| easy,nonn,changed
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001
Additional comments from Michael Somos, Feb 24, 2002
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