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%I M3574 #117 Apr 15 2024 12:59:48
%S 1,1,4,21,126,818,5594,39693,289510,2157150,16348960,125642146,
%T 976789620,7668465964,60708178054,484093913917,3884724864390,
%U 31348290348086,254225828706248,2070856216759478,16936016649259364
%N Number of blobs with 2n+1 edges.
%C a(n) is the number of ways to dissect a convex (2n+2)-gon with non-crossing diagonals so that no (2m+1)-gons (m>0) appear. - _Len Smiley_
%C a(n) is the number of plane trees with 2n+1 leaves and all non-leaves having an odd number > 1 of children. - _Jordan Tirrell_, Jun 09 2017
%C a(n) is the number of noncrossing cacti with n+1 nodes. See A361242. - _Andrew Howroyd_, Mar 07 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A003168/b003168.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from T. D. Noe)
%H Thomas H. Bertschinger, Joseph Slote, Olivia Claire Spencer, and Samuel Vinitsky, <a href="https://joeslote.com/documents/origami_undergrad_thesis.pdf">The Mathematics of Origami</a>, Undergrad Thesis, Carleton College (2016).
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
%H L. Carlitz, <a href="http://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
%H Frédéric Chapoton and Philippe Nadeau, <a href="http://www.mat.univie.ac.at/~slc/wpapers/FPSAC2017/37%20Chapoton%20Nadeau.pdf">Combinatorics of the categories of noncrossing partitions</a>, Séminaire Lotharingien de Combinatoire 78B (2017), Article #37.
%H Michael Drmota, Anna de Mier, and Marc Noy, <a href="http://www.dmg.tuwien.ac.at/drmota/noncrossingFinal4.pdf">Extremal statistics on non-crossing configurations</a>, Discrete Math. 327 (2014), 103--117. MR3192420. See p. 116, B_b(z). - N. J. A. Sloane, May 18 2014
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=415">Encyclopedia of Combinatorial Structures 415</a>
%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://arxiv.org/abs/2106.08257">Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra</a>, arXiv:2106.08257 [math.CO], 2021-2022.
%H Ronald C. Read, <a href="https://doi.org/10.1007/BF02188172">On the enumeration of a class of plane multigraphs</a>, Aequat. Math. 31 (1986) No. 1, 47-63.
%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/vsd2.html">Even-gon reference</a>
%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 30.
%F a(n) = Sum_{k=1..n} binomial(n, k)*binomial(2*n+k, k-1)/n.
%F G.f.: A(x) = Sum_{n>=0} a(n)*x^(2*n+1) satisfies (A-2*A^3)/(1-A^2)=x. - _Len Smiley_.
%F D-finite with recurrence 4*n*(2*n + 1)*(17*n - 22)*a(n) = (1207*n^3 - 2769*n^2 + 1850*n - 360)*a(n - 1) - 2*(17*n - 5)*(n - 2)*(2*n - 3)*a(n - 2). - _Vladeta Jovovic_, Jul 16 2004
%F G.f.: A(x) = 1/(1-G003169(x)) where G003169(x) is the g.f. of A003169. - _Paul D. Hanna_, Nov 17 2004
%F a(n) = JacobiP(n-1,1,n+1,3)/n for n > 0. - _Mark van Hoeij_, Jun 02 2010
%F a(n) = (1/(2*n+1))*Sum_{j=0..n} (-1)^j*2^(n-j)*binomial(2*n+1,j)*binomial(3*n-j,2*n). - _Vladimir Kruchinin_, Dec 24 2010
%F From _Gary W. Adamson_, Jul 08 2011: (Start)
%F a(n) = upper left term in M^n, M = the production matrix:
%F 1, 1
%F 3, 3, 1
%F 5, 5, 3, 1
%F 7, 7, 5, 3, 1
%F 9, 9, 7, 5, 3, 1
%F ... (End)
%F a(n) ~ sqrt(14+66/sqrt(17)) * (71+17*sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(4*n+4)). - _Vaclav Kotesovec_, Jul 01 2015
%F From _Peter Bala_, Oct 05 2015: (Start)
%F a(n) = (1/n) * Sum_{i = 0..n} 2^(n-i-1)*binomial(2*n,i)* binomial(n,i+1).
%F O.g.f. = 1 + series reversion( x/((1 + 2*x)*(1 + x)^2) ).
%F Logarithmically differentiating the modified g.f. 1 + 4*x + 21*x^2 + 126*x^3 + 818*x^4 + ... gives the o.g.f. for A114496, apart from the initial term. (End)
%F G.f.: A(x) satisfies A = 1 + x*A^3/(1-x*A^2). - _Jordan Tirrell_, Jun 09 2017
%F a(n) = A100327(n)/2 for n>=1. - _Peter Luschny_, Jun 10 2017
%e a(2)=4 because we may place exactly one diagonal in 3 ways (forming 2 quadrilaterals), or not place any (leaving 1 hexagon).
%p Order := 40; solve(series((A-2*A^3)/(1-A^2),A)=x,A);
%p A003168 := n -> `if`(n=0,1,A100327(n)/2): seq(A003168(n),n=0..20); # _Peter Luschny_, Jun 10 2017
%t a[0] = 1; a[n_] = (2^(-n-1)*(3n)!* Hypergeometric2F1[-1-2n, -2n, -3n, -1])/((2n+1)* n!*(2n)!); Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 25 2011, after _Vladimir Kruchinin_ *)
%o (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^3)/(1-x^2)+O(x^(2*n+2))),2*n+1))
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x*A)/(1-x*A)^2); sum(k=0,n,polcoeff(A^(n-k),k))} \\ _Paul D. Hanna_, Nov 17 2004
%o (PARI) seq(n) = Vec( 1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n)) ) \\ _Andrew Howroyd_, Mar 07 2023
%o (Haskell)
%o import Data.List (transpose)
%o a003168 0 = 1
%o a003168 n = sum (zipWith (*)
%o (tail $ a007318_tabl !! n)
%o ((transpose $ take (3*n+1) a007318_tabl) !! (2*n+1)))
%o `div` fromIntegral n
%o -- _Reinhard Zumkeller_, Oct 27 2013
%Y Cf. A049124 (no 2m-gons).
%Y Cf. A003169, A100327, A114496, A007318, A361242.
%Y Row sums of A102537, A243662.
%K nonn,easy,nice,changed
%O 0,3
%A _N. J. A. Sloane_
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