%I #63 Sep 08 2022 08:45:48

%S 1,1,3,7,22,65,213,693,2352,8034,28014,98505,350548,1256827,4542395,

%T 16517631,60417708,222087320,820099720,3040555978,11314532376,

%U 42243332130,158196980682,594075563613,2236627194858,8440468925400

%N Number of possible paths to each node that lies along the edge of a cut 4-nomial tree, that is rooted one unit from the cut.

%C This is the third member of an infinite series of infinite series, the first two being the Catalan and Motzkin integers. The Catalan numbers lie on the edge of cut 2-nomial trees, Motzkin integers on the edge of cut 3-nomial trees.

%C a(n) is the number of increasing unary-binary trees with associated permutation that avoids 213. For more information about increasing unary-binary trees with an associated permutation, see A245888. - _Manda Riehl_, Aug 07 2014

%C Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - _David Nguyen_, Dec 16 2016

%H G. C. Greubel, <a href="/A166135/b166135.txt">Table of n, a(n) for n = 1..1000</a>

%H Cyril Banderier, Christian Krattenthaler, Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen, and Michael Wallner, <a href="http://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 Jean-Luc Baril and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/knight.pdf">Knight's paths towards Catalan numbers</a>, Univ. Bourgogne Franche-Comté (2022).

%H Jérémie Bettinelli, Éric Fusy, Cécile Mailler, and Lucas Randazzo, <a href="https://arxiv.org/abs/1611.01478">A bijective study of Basketball walks</a>, arXiv:1611.01478 [math.CO], 2016.

%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.

%H Rick Jarosh, <a href="http://www.jarosh.net/psi/PsiFourOne.jpg">Illustration of 4-nomial graph</a> The series is the one at the top.

%H Rick Jarosh, <a href="http://www.jarosh.net/psi/PsiFourOneOne.txt">First 4096 terms of the series in a text file.</a>

%H Rick Jarosh, <a href="http://www.jarosh.net/psi/PsiM11.txt">Illustrates the sequence in context.</a> The above reference gives the first 16 terms of the first 128 sequences in the family, of which this sequence is the third, the first being the Catalan numbers, the second the Motzkin integers, the fourth A104632.

%F a(n) = ((36*n+18)*A092765(n) + (11*n+9)*A092765(n+1))/(2*(5*n+3)*(2*n+3)) (based on guessed recurrence). - _Mark van Hoeij_, Jul 14 2010

%F A(x) satisfies A(x)+A(x)^2 = A000108(x)-1, a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1) * C(2*n,n-k)*C(2*k-2,k-1). - _Vladimir Kruchinin_, May 12 2012

%F G.f.: (sqrt((2 - 2*sqrt(1-4*x) - 3*x)/x) - 1)/2. - _Benedict W. J. Irwin_, Sep 24 2016

%F a(n) ~ 4^n/(sqrt(5*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 25 2016

%F Conjecture: 2*n*(2*n+1)*a(n) + (17*n^2-53*n+24)*a(n-1) + 6*(-13*n^2+43*n-36)*a(n-2) - 108*(2*n-5)*(n-3)*a(n-3) = 0. - _R. J. Mathar_, Oct 08 2016

%F a(n) = (1/n)*Sum_{k=0..n} binomial(n,k)*binomial(n,2*n-3*k-1). - _David Nguyen_, Dec 31 2016

%F From _Alexander Burstein_, Dec 12 2019: (Begin)

%F 1 + x*A(x) = 1/C(-x*C(x)^2), where C(x) is the g.f. of A000108.

%F F(x) = x*(1+x*A(x)) = x/C(-x*C(x)^2) is a pseudo-involution, i.e., the series reversion of x*(1 + x*A(x)) is x*(1 - x*A(-x)).

%F The B-sequence of F(x) is A069271, i.e., F(x) = x + x*F(x)*A069271(x*F(x)). (End)

%p seq( add(binomial(n,k)*binomial(n,2*n-3*k-1), k=0..n)/n, n=1..30); # _G. C. Greubel_, Dec 12 2019

%t Rest[CoefficientList[Series[(Sqrt[(2-2Sqrt[1-4x]-3x)/x]-1)/2, {x, 0, 30}],x]] (* _Benedict W. J. Irwin_, Sep 24 2016 *)

%o (PARI) vector(30, n, sum(k=0,n, binomial(n,k)*binomial(n,2*n-3*k-1))/n ) \\ _G. C. Greubel_, Dec 12 2019

%o (Magma) [(&+[Binomial(n,k)*Binomial(n,2*n-3*k-1): k in [0..n]])/n : n in [1..30]]; // _G. C. Greubel_, Dec 12 2019

%o (Sage) [sum(binomial(n,k)*binomial(n,2*n-3*k-1) for k in (0..n))/n for n in (1..30)] # _G. C. Greubel_, Dec 12 2019

%Y A055113 is the third sequence from the top of the graph illustrated above.

%Y Cf. A000108, A001006, A069271, A092765, A126120.

%K nonn

%O 1,3

%A Rick Jarosh (rick(AT)jarosh.net), Oct 08 2009