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A047749 If n = 2m then a(n) = binomial(3m,m)/(2m+1); if n=2m+1 then a(n) = binomial(3m+1,m+1)/(2m+1). 33
1, 1, 1, 2, 3, 7, 12, 30, 55, 143, 273, 728, 1428, 3876, 7752, 21318, 43263, 120175, 246675, 690690, 1430715, 4032015, 8414640, 23841480, 50067108, 142498692, 300830572, 859515920, 1822766520, 5225264024, 11124755664, 31983672534 (list; graph; refs; listen; history; text; internal format)
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

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

|..............|

\/....\|/....\/ .. - David Callan, Aug 22 2014

a(n) is the number of symmetric ternary trees having n internal nodes. - Emeric Deutsch, Oct 28 2014

a(n) is the number of symmetric non-crossing rooted trees having n edges. - Emeric Deutsch, Oct 28 2014

a(n) is the number of symmetric even trees having 2n edges. - Emeric Deutsch, Oct 28 2014

a(n) is the number of symmetric diagonally convex directed polyominoes having n diagonals. - Emeric Deutsch, Oct 28 2014

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

LINKS

Table of n, a(n) for n=0..31.

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.

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 C├ędric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.

Hassen Cheriha, Yousra Gati, Vladimir Petrov Kostov, Descartes' rule of signs, Rolle's theorem and sequences of admissible pairs, arXiv:1805.04261 [math.CA], 2018.

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.CO/0610234. [Theorem 3.5]

E. Deutsch, Another Path to Generalized Catalan Numbers:Problem 10751, Amer. Math. Monthly, 108 (Nov., 2001), 872-873.

E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Anthony Zaleski, 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-2x^2)/(1-x)^3 (ignoring signs). - Ralf Stephan, Mar 22 2004

G.f.: (4/(3x))(sin((1/3)*asin(sqrt(27x^2/4))))^2+(2/sqrt(3x^2))*sin((1/3)*asin(sqrt(27x^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

a(n) = 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, ...

  ...

- Gary W. Adamson, Jul 14 2011

Conjecture D-finite with recurrence: 8*n*(n+1)*a(n) + 36*n*(n-2)*a(n-1) - 6*(9n^2-18n+14)*a(n-2) - 27*(3n-7)*(3n-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

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

A047749 := proc(m) if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; x := m/2; RETURN((3*x)!/(x!*(2*x+1)!)); end;

A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end;

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 *)

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) [Round((1+(-1)^n)*Gamma(3*n/2+1)/(Gamma(n/2+1)*Factorial(n+1)) + (1-(-1)^n)*Gamma((3*n+1)/2)/(Gamma((n+3)/2)*Factorial(n)))/2: n in [0..35]]; // G. C. Greubel, Jul 07 2019

CROSSREFS

Cf. A001764.

Cf. A006013 is the odd-indexed terms of this sequence.

Sequence in context: A297438 A111759 A305751 * A134565 A300749 A100982

Adjacent sequences:  A047746 A047747 A047748 * A047750 A047751 A047752

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 5 17:16 EST 2020. Contains 338954 sequences. (Running on oeis4.)