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A005159
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3^n*Catalan(n).
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16
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1, 3, 18, 135, 1134, 10206, 96228, 938223, 9382230, 95698746, 991787004, 10413763542, 110546105292, 1184422556700, 12791763612360, 139110429284415, 1522031755700070, 16742349312700770, 185047018719324300, 2054021907784499730
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Total number of vertices in rooted planar maps with n edges.
Number of blossom trees with n inner vertices.
The number of rooted n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005
Hankel transform is 3^(n+n^2)=A053764(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2007
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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/AoifeTh
L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
V. A. Liskovets, A pattern of asymptotic vertex valency distributions in planar maps. J. Combin. Th., B75 (1999), 116-133.
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
| T. D. Noe, Table of n, a(n) for n=0..100
M. Bousquet-Melou, Limit laws for embedded trees
G. 't Hooft, Counting planar diagrams with various restrictions, Nucl. Phys. B538 (1999), 389-410.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.
G. Schaeffer and P. Zinn-Justin, On the asymptotic number of plane curves and alternating knots
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FORMULA
| G.f.: 2/(1+sqrt(1-12x)).
With offset 1 : a(1)=1, a(n)=3*sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004
G.f.: c(3*x) with c(x) the o.g.f. of A000108 (Catalan).
a(n) = upper left term in M^n, M = the infinite square production matrix:
3, 3, 0, 0, 0, 0,...
3, 3, 3, 0, 0, 0,...
3, 3, 3, 3, 0, 0,...
3, 3, 3, 3, 3, 0,...
3, 3, 3, 3, 3, 3,...
...
- Gary W. Adamson, Jul 12 2011
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MAPLE
| A005159_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od; convert(a, list)end: A005159_list(19); # Peter Luschny, May 19 2011
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MATHEMATICA
| InverseSeries[Series[y-3*y^2, {y, 0, 24}], x] (* then A(x)=y(x)/x *) - Len Smiley Apr 07 2000
Table[3^n CatalanNumber[n], {n, 0, 30}] (* From Harvey P. Dale, May 18 2011 *)
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CROSSREFS
| Cf. A000108, A025226.
Limit of array A102994.
Sequence in context: A168072 A095776 A114178 * A151383 A177406 A118970
Adjacent sequences: A005156 A005157 A005158 * A005160 A005161 A005162
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Valery A. Liskovets (liskov(AT)im.bas-net.by)
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