OFFSET
0,4
COMMENTS
Hankel transform appears to be a signed aerated version of A059492. - Paul Barry, Apr 16 2008
Row sums of inverse Riordan array (1, x*(1-x^2))^(-1). - Paul Barry, Apr 16 2008
a(n) is the number of permutations of length n avoiding 213 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
From David Callan, Aug 22 2014: (Start)
a(n) is the number of ordered trees (A000108) with n vertices in which every non-root non-leaf vertex has exactly one leaf child (no restriction on its non-leaf children). For example, a(4) counts the 3 trees
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(End)
From Emeric Deutsch, Oct 28 2014: (Start)
a(n) is the number of symmetric ternary trees having n internal nodes.
a(n) is the number of symmetric non-crossing rooted trees having n edges.
a(n) is the number of symmetric even trees having 2n edges.
a(n) is the number of symmetric diagonally convex directed polyominoes having n diagonals.
(End)
For the above 4 items see the Deutsch-Feretic-Noy reference.
a(n) is also the number of self-dual labeled non-crossing trees with n edges. See my paper in the links section. - Nikos Apostolakis, Jun 11 2019
Number of achiral polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. An achiral polyomino is identical to its reflection. - Robert A. Russell, Jan 20 2024
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
Nikos Apostolakis, Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections arXiv:1804.01214 [math.CO], 2018.
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Theorem 4.4.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67.
M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.
Hassen Cheriha, Yousra Gati, and Vladimir Petrov Kostov, Descartes' rule of signs, Rolle's theorem and sequences of admissible pairs, arXiv:1805.04261 [math.CA], 2018.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.
S. J. Cyvin et al., Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes, J. Molec. Struct., 445 (1998), 127-137.
Alexander Burstein, Sergi Elizalde, and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. [Theorem 3.5]
Emeric Deutsch, Another Path to Generalized Catalan Numbers:Problem 10751, Amer. Math. Monthly, 108 (Nov., 2001), 872-873.
Emeric Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
Anthony Zaleski and Doron Zeilberger, On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts, arXiv:1712.10072 [math.CO], 2017.
FORMULA
G.f. is 1+Z, where Z satisfies x*Z^3 + (3*x-2)*Z^2 + (3*x-1)*Z + x = 0. Equivalently, the g.f. Y satisfies x*Y^3 - 2*Y^2 + 3*Y - 1 = 0. - Vladeta Jovovic, Dec 06 2002
Reversion of g.f. (x-2*x^2)/(1-x)^3 (ignoring signs). - Ralf Stephan, Mar 22 2004
G.f.: (4/(3*x))*(sin((1/3)*asin(sqrt(27*x^2/4))))^2 +(2/sqrt(3*x^2))*sin((1/3)*asin(sqrt(27*x^2/4))). - Paul Barry, Nov 08 2006
G.f.: 1/(1-2*sin(asin(3*sqrt(3)*x/2)/3)/sqrt(3)). - Paul Barry, Apr 16 2008
From Paul D. Hanna, Sep 20 2009: (Start)
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x);
also, A(x)*A(-x) = B(x^2) where B(x) = 1 + x*B(x)^3 = g.f. of A001764.
(End)
G.f.: 1/(1-C(x)) where C(x) = Reverse(x-x^3) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 + ... (cf. A001764). - Joerg Arndt, Apr 16 2011
G.f.: G(z^2)+z*G(z^2)^2, where G(z) = 1 + z*G(z)^3, the generating function for A001764. - Robert A. Russell, Jan 26 2024
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) is the upper left term in M^n, M = the infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
0, 0, 1, 0, 0, 0, ...
1, 1, 0, 1, 0, 0, ...
0, 0, 1, 0, 1, 0, ...
1, 1, 0, 1, 0, 1, ...
... (End)
Conjecture D-finite with recurrence: 8*n*(n+1)*a(n) + 36*n*(n-2)*a(n-1) - 6*(9*n^2-18*n+14)*a(n-2) - 27*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Dec 19 2011
0 = a(n)*(+7308954*a(n+2) - 16659999*a(n+3) - 4854519*a(n+4) + 6201838*a(n+5)) + a(n+1)*(-406053*a(n+2) - 1627560*a(n+3) + 1683538*a(n+4) - 245747*a(n+5)) + a(n+2)*(+45117*a(n+2) + 235870*a(n+3) + 173953*a(n+4) - 484295*a(n+5)) + a(n+3)*(-41820*a(n+3) - 50184*a(n+4) + 22304*a(n+5)) for all n in Z if a(-1) = -2/3. - Michael Somos, Oct 29 2014
a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} (-1)^i * a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 28 2021
a(n) = binomial(A032766(n),floor((n+1)/2))/(2*floor(n/2)+1). - Miko Labalan, Nov 28 2023
a(n) = 2*A005036(n) - A005034(n) = A005034(n) - 2*A369315(n) = A005036(n) - A369315(n). - Robert A. Russell, Jan 20 2024
From Robert A. Russell, Mar 20 2024: (Start)
a(n) = U(n) in the Beineke and Pippert link.
G.f.: E(1)(t*E(3)(t^2)) (second entry in Table 1), where E(d)(t) is defined in formula 3 of Hering link. (End)
From Robert A. Russell, Jul 15 2024: (Start)
a(2m) = A001764(m) ~ (3^3/2^2)^m*sqrt(3/(2*Pi*(2*m)^3)).
a(n+2)/a(n) ~ 27/4; a(2m+1)/a(2m) ~ 3; a(2m)/a(2m-1) ~ 9/4. (End)
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 12*x^6 + 30*x^7 + 55*x^8 + ...
MAPLE
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], SeriesCoefficient[ InverseSeries[ Series[ (x + 2 x^2) / (1 + x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 29 2014 *)
Table[If[OddQ[n], 2Binomial[(3n-1)/2, (n-1)/2], Binomial[3n/2, n/2]]/(n+1), {n, 0, 40}] (* Robert A. Russell, Jan 19 2024 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2*subst(A, x, -x+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Sep 20 2009
(PARI) x='x+O('x^66);
C(x)=serreverse(x-x^3); /* =x+x^3+3*x^5+12*x^7+55*x^9 +..., cf. A001764 */
s=1/(1-C(x)); /* g.f. */
Vec(s) /* Joerg Arndt, Apr 16 2011 */
(Sage)
def A047749_list(n) :
D = [0]*n; D[1] = 1
R = []; b = False; h = 1
for i in range(n) :
for k in (1..h) :
D[k] = D[k] + D[k-1]
R.append(D[h])
if b : h += 1
b = not b
return R
A047749_list(35) # Peter Luschny, May 03 2012
(Sage) [1]+[((1+(-1)^n)*binomial(3*n/2, n/2)/(n+1) + (1-(-1)^n)* binomial((3*n-1)/2, (n+1)/2)/n)/2 for n in (1..35)] # G. C. Greubel, Jul 07 2019
(Magma) G:=Gamma; [Round((1+(-1)^n)*G(3*n/2+1)/(G(n/2+1)*Factorial(n+1)) + (1-(-1)^n)*G((3*n+1)/2)/(G((n+3)/2)*Factorial(n)))/2: n in [0..35]]; // G. C. Greubel, Jul 07 2019
(Python)
from math import comb
def A047749(n): return comb(n+(a:=n>>1), a+(b:=n&1))//(n+1-b) # Chai Wah Wu, Jul 30 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved