|
%I #207 Feb 05 2024 10:51:51
%S 1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,
%T 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,
%U 477638700,0,1767263190,0,6564120420,0,24466267020,0,91482563640,0,343059613650,0
%N Catalan numbers (A000108) interpolated with 0's.
%C Inverse binomial transform of A001006.
%C The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
%C 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
%C 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
%C Essentially the same as A097331. - _R. J. Mathar_, Jun 15 2008
%C Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010
%C -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
%C 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
%C 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
%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.
%H G. C. Greubel, <a href="/A126120/b126120.txt">Table of n, a(n) for n = 0..1000</a>
%H V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.
%H Martin Aigner, <a href="http://dx.doi.org/10.1007/978-88-470-2107-5_15">Catalan and other numbers: a recurrent theme</a>, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv:1609.06473 [math.CO], 2016.
%H Radica Bojicic, Marko D. Petkovic and Paul Barry, <a href="http://arxiv.org/abs/1112.1656">Hankel transform of a sequence obtained by series reversion II-aerating transforms</a>, arXiv:1112.1656 [math.CO], 2011.
%H Colin Defant, <a href="https://arxiv.org/abs/2004.11367">Troupes, Cumulants, and Stack-Sorting</a>, arXiv:2004.11367 [math.CO], 2020.
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv:1110.6638 [math.NT], 2011.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://dspace.mit.edu/handle/1721.1/64701">HyperellipticCurves, L-Polynomials, and Random Matrices</a>. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)
%H S. Mizera, <a href="https://arxiv.org/abs/1706.08527">Combinatorics and Topology of Kawai-Lewellen-Tye Relations</a>, arXiv:1706.08527 [hep-th], 2017.
%H E. Rowland, <a href="https://doi.org/10.1016/j.jcta.2010.03.004">Pattern avoidance in binary trees</a>, J. Comb. Theory A 117 (6) (2010) 741-758, Sec. 3.1.
%H Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2015">Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths</a>, arXiv:1305.2015 [math.CO], 2013.
%H Y. Wang and Z.-H. Zhang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Wang/wang21.html">Combinatorics of Generalized Motzkin Numbers</a>, J. Int. Seq. 18 (2015) # 15.2.4.
%F a(2*n) = A000108(n), a(2*n+1) = 0.
%F a(n) = A053121(n,0).
%F (1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - _Andrew V. Sutherland_, Feb 29 2008
%F 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
%F G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - _Vladimir Kruchinin_, Feb 18 2011
%F E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - _Benjamin Phillabaum_, Mar 07 2011
%F 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
%F a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x)))/(2*Pi). - _Peter Luschny_, Sep 11 2011
%F 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
%F 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)
%F 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
%F a(n) = n!*[x^n]hypergeom([],[2],x^2). - _Peter Luschny_, Jan 31 2015
%F a(n) = 2^n*hypergeom([3/2,-n],[3],2). - _Peter Luschny_, Feb 03 2015
%F 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
%F D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Mar 21 2021
%F From _Peter Bala_, Feb 03 2024: (Start)
%F a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).
%F G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^2 = 1/(1 - 2*x) * c(-x/(1 - 2*x))^2 = c(x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%e G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
%e From _Gus Wiseman_, Nov 14 2022: (Start)
%e The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
%e o . (oo) . ((oo)o) . (((oo)o)o) . ((((oo)o)o)o)
%e (o(oo)) ((o(oo))o) (((o(oo))o)o)
%e ((oo)(oo)) (((oo)(oo))o)
%e (o((oo)o)) (((oo)o)(oo))
%e (o(o(oo))) ((o((oo)o))o)
%e ((o(o(oo)))o)
%e ((o(oo))(oo))
%e ((oo)((oo)o))
%e ((oo)(o(oo)))
%e (o(((oo)o)o))
%e (o((o(oo))o))
%e (o((oo)(oo)))
%e (o(o((oo)o)))
%e (o(o(o(oo))))
%e (End)
%p 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
%p 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
%p 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
%p seq(n!*coeff(series(hypergeom([],[2],x^2),x,n+2),x,n),n=0..45); # _Peter Luschny_, Jan 31 2015
%p # Using function CompInv from A357588.
%p CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # _Peter Luschny_, Oct 07 2022
%t a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Sep 10 2012 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* _Michael Somos_, Mar 19 2014 *)
%t bot[n_]:=If[n==1,{{}},Join@@Table[Tuples[bot/@c],{c,Table[{k,n-k-1},{k,n-1}]}]];
%t Table[Length[bot[n]],{n,10}] (* _Gus Wiseman_, Nov 14 2022 *)
%o (Sage)
%o def A126120_list(n) :
%o D = [0]*(n+2); D[1] = 1
%o b = True; h = 2; R = []
%o for i in range(2*n-1) :
%o if b :
%o for k in range(h,0,-1) : D[k] -= D[k-1]
%o h += 1; R.append(abs(D[1]))
%o else :
%o for k in range(1,h, 1) : D[k] += D[k+1]
%o b = not b
%o return R
%o A126120_list(46) # _Peter Luschny_, Jun 03 2012
%o (Magma) &cat [[Catalan(n), 0]: n in [0..30]]; // _Vincenzo Librandi_, Jul 28 2016
%Y Cf. A000108, A036039, A036040, A097610, A127671, A180874, A238390, A263916.
%Y Cf. A126216.
%Y The unordered version is A001190, ranked by A111299.
%Y These trees (ordered binary rooted) are ranked by A358375.
%Y Cf. A000081, A001263, A005043, A032027, A063895, A245824.
%K nonn,easy
%O 0,5
%A _Philippe Deléham_, Mar 06 2007
%E An erroneous comment removed by _Tom Copeland_, Jul 23 2016
|
