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A126120 Catalan numbers (A000108) interpolated with 0's. 50
1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Inverse binomial transform of A001006.

The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].

Counts returning walks (excursions) of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland, Feb 29 2008

Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008

Essentially the same as A097331. - R. J. Mathar, Jun 15 2008

Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010

-a(n-1), with a(-1):=0, n>=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - Wolfdieter Lang, Nov 04 2011

See A180874 (also A238390 and A097610) and A263916 for relations to the general Bell A036040, cycle index A036039, and cumulant expansion polynomials A127671 through the Faber polynomials. - Tom Copeland, Jan 26 2016

A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - Tom Copeland, Dec 09 2019

REFERENCES

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.

Martin Aigner, Catalan and other numbers: a recurrent theme, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Radica Bojicic, Marko D. Petkovic and Paul Barry, Hankel transform of a sequence obtained by series reversion II-aerating transforms, arXiv:1112.1656 [math.CO], 2011.

Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.

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.

Kiran S. Kedlaya and Andrew V. Sutherland, HyperellipticCurves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)

S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, arXiv:1706.08527 [hep-th], 2017.

E. Rowland, Pattern avoidance in binary trees, J. Comb. Theory A 117 (6) (2010) 741-758, Sec. 3.1.

Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv:1305.2015 [math.CO], 2013.

Y. Wang and Z.-H. Zhang, Combinatorics of Generalized Motzkin Numbers, J. Int. Seq. 18 (2015) # 15.2.4.

FORMULA

a(2*n) = A000108(n), a(2*n+1) = 0.

a(n) = A053121(n,0).

(1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - Andrew V. Sutherland, Feb 29 2008

G.f.: (1 - sqrt(1 - 4*x^2)) / (2*x^2) = 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-...(continued fraction). - Philippe Deléham, Nov 24 2009

G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - Vladimir Kruchinin, Feb 18 2011

E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - Benjamin Phillabaum, Mar 07 2011

Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - Benjamin Phillabaum, Mar 07 2011

a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x)))/(2*Pi). - Peter Luschny, Sep 11 2011

E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013

G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013 (warning: this is not the g.f. of this sequence, R. J. Mathar, Sep 23 2021)

G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014

a(n) = n!*[x^n]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015

a(n) = 2^n*hypergeom([3/2,-n],[3],2). - Peter Luschny, Feb 03 2015

a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - Ilya Gutkovskiy, Jul 23 2016

D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Mar 21 2021

EXAMPLE

G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...

MAPLE

with(combstruct): grammar := { BB = Sequence(Prod(a, BB, b)), a = Atom, b = Atom }: seq(count([BB, grammar], size=n), n=0..47); # Zerinvary Lajos, Apr 25 2007

BB := {E=Prod(Z, Z), S=Union(Epsilon, Prod(S, S, E))}: ZL:=[S, BB, unlabeled]: seq(count(ZL, size=n), n=0..45); # Zerinvary Lajos, Apr 22 2007

BB := [T, {T=Prod(Z, Z, Z, F, F), F=Sequence(B), B=Prod(F, Z, Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n> 0. # Zerinvary Lajos, Apr 22 2007

seq(n!*coeff(series(hypergeom([], [2], x^2), x, n+2), x, n), n=0..45); # Peter Luschny, Jan 31 2015

MATHEMATICA

a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Sep 10 2012 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* Michael Somos, Mar 19 2014 *)

PROG

(Sage)

def A126120_list(n) :

    D = [0]*(n+2); D[1] = 1

    b = True; h = 2; R = []

    for i in range(2*n-1) :

        if b :

            for k in range(h, 0, -1) : D[k] -= D[k-1]

            h += 1; R.append(abs(D[1]))

        else :

            for k in range(1, h, 1) : D[k] += D[k+1]

        b = not b

    return R

A126120_list(46) # Peter Luschny, Jun 03 2012

(Magma) &cat [[Catalan(n), 0]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2016

CROSSREFS

Cf. A000108.

Cf. A036039, A036040,  A097610, A127671, A180874, A238390, A263916.

Cf. A126216.

Sequence in context: A210628 A090192 A097331 * A260330 A094032 A261044

Adjacent sequences:  A126117 A126118 A126119 * A126121 A126122 A126123

KEYWORD

nonn

AUTHOR

Philippe Deléham, Mar 06 2007

EXTENSIONS

An erroneous comment removed by Tom Copeland, Jul 23 2016

STATUS

approved

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Last modified October 5 17:31 EDT 2022. Contains 357259 sequences. (Running on oeis4.)