This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A126120 #231 May 08 2025 16:37:28
%S A126120 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 A126120 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,
%U A126120 477638700,0,1767263190,0,6564120420,0,24466267020,0,91482563640,0,343059613650,0
%N A126120 Catalan numbers (A000108) interpolated with 0's.
%C A126120 Inverse binomial transform of A001006.
%C A126120 The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
%C A126120 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 A126120 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 A126120 Essentially the same as A097331. - _R. J. Mathar_, Jun 15 2008
%C A126120 Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010
%C A126120 -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 A126120 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 A126120 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 A126120 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.
%H A126120 G. C. Greubel, <a href="/A126120/b126120.txt">Table of n, a(n) for n = 0..1000</a>
%H A126120 V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.
%H A126120 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 A126120 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 A126120 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 A126120 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 A126120 Colin Defant, <a href="https://arxiv.org/abs/2004.11367">Troupes, Cumulants, and Stack-Sorting</a>, arXiv:2004.11367 [math.CO], 2020.
%H A126120 Isaac DeJager, Madeleine Naquin, and 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 A126120 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 A126120 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 A126120 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 A126120 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 A126120 Eric 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 A126120 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 A126120 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.
%H A126120 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 A126120 a(2*n) = A000108(n), a(2*n+1) = 0.
%F A126120 a(n) = A053121(n,0).
%F A126120 (1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - _Andrew V. Sutherland_, Feb 29 2008
%F A126120 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 A126120 G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - _Vladimir Kruchinin_, Feb 18 2011
%F A126120 E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - _Benjamin Phillabaum_, Mar 07 2011
%F A126120 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 A126120 a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x))/(2*Pi) dx. - _Peter Luschny_, Sep 11 2011
%F A126120 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 A126120 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 A126120 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 A126120 a(n) = n!*[x^n]hypergeom([],[2],x^2). - _Peter Luschny_, Jan 31 2015
%F A126120 a(n) = 2^n*hypergeom([3/2,-n],[3],2). - _Peter Luschny_, Feb 03 2015
%F A126120 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 A126120 D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Mar 21 2021
%F A126120 From _Peter Bala_, Feb 03 2024: (Start)
%F A126120 a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).
%F A126120 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 A126120 G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
%e A126120 From _Gus Wiseman_, Nov 14 2022: (Start)
%e A126120 The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
%e A126120 o . (oo) . ((oo)o) . (((oo)o)o) . ((((oo)o)o)o)
%e A126120 (o(oo)) ((o(oo))o) (((o(oo))o)o)
%e A126120 ((oo)(oo)) (((oo)(oo))o)
%e A126120 (o((oo)o)) (((oo)o)(oo))
%e A126120 (o(o(oo))) ((o((oo)o))o)
%e A126120 ((o(o(oo)))o)
%e A126120 ((o(oo))(oo))
%e A126120 ((oo)((oo)o))
%e A126120 ((oo)(o(oo)))
%e A126120 (o(((oo)o)o))
%e A126120 (o((o(oo))o))
%e A126120 (o((oo)(oo)))
%e A126120 (o(o((oo)o)))
%e A126120 (o(o(o(oo))))
%e A126120 (End)
%p A126120 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 A126120 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 A126120 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 A126120 seq(n!*coeff(series(hypergeom([],[2],x^2),x,n+2),x,n),n=0..45); # _Peter Luschny_, Jan 31 2015
%p A126120 # Using function CompInv from A357588.
%p A126120 CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # _Peter Luschny_, Oct 07 2022
%t A126120 a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Sep 10 2012 *)
%t A126120 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* _Michael Somos_, Mar 19 2014 *)
%t A126120 bot[n_]:=If[n==1,{{}},Join@@Table[Tuples[bot/@c],{c,Table[{k,n-k-1},{k,n-1}]}]];
%t A126120 Table[Length[bot[n]],{n,10}] (* _Gus Wiseman_, Nov 14 2022 *)
%t A126120 Riffle[CatalanNumber[Range[0,50]],0,{2,-1,2}] (* _Harvey P. Dale_, May 28 2024 *)
%o A126120 (Sage)
%o A126120 def A126120_list(n) :
%o A126120 D = [0]*(n+2); D[1] = 1
%o A126120 b = True; h = 2; R = []
%o A126120 for i in range(2*n-1) :
%o A126120 if b :
%o A126120 for k in range(h,0,-1) : D[k] -= D[k-1]
%o A126120 h += 1; R.append(abs(D[1]))
%o A126120 else :
%o A126120 for k in range(1,h, 1) : D[k] += D[k+1]
%o A126120 b = not b
%o A126120 return R
%o A126120 A126120_list(46) # _Peter Luschny_, Jun 03 2012
%o A126120 (Magma) &cat [[Catalan(n), 0]: n in [0..30]]; // _Vincenzo Librandi_, Jul 28 2016
%o A126120 (Python)
%o A126120 from math import comb
%o A126120 def A126120(n): return 0 if n&1 else comb(n,m:=n>>1)//(m+1) # _Chai Wah Wu_, Apr 22 2024
%Y A126120 Cf. A000108, A036039, A036040, A097610, A127671, A180874, A238390, A263916.
%Y A126120 Cf. A126216.
%Y A126120 The unordered version is A001190, ranked by A111299.
%Y A126120 These trees (ordered binary rooted) are ranked by A358375.
%Y A126120 Cf. A000081, A001263, A005043, A032027, A063895, A245824.
%K A126120 nonn,easy
%O A126120 0,5
%A A126120 _Philippe Deléham_, Mar 06 2007
%E A126120 An erroneous comment removed by _Tom Copeland_, Jul 23 2016