A166135 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.

1, 1, 3, 7, 22, 65, 213, 693, 2352, 8034, 28014, 98505, 350548, 1256827, 4542395, 16517631, 60417708, 222087320, 820099720, 3040555978, 11314532376, 42243332130, 158196980682, 594075563613, 2236627194858, 8440468925400

Offset

1,3

Comments

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.

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cyril Banderier, Christian Krattenthaler, Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen, and Michael Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).

Jérémie Bettinelli, Éric Fusy, Cécile Mailler, and Lucas Randazzo, A bijective study of Basketball walks, arXiv:1611.01478 [math.CO], 2016.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Rick Jarosh, Illustration of 4-nomial graph The series is the one at the top.

Rick Jarosh, First 4096 terms of the series in a text file.

Rick Jarosh, Illustrates the sequence in context. 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.

Formula

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

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

G.f.: (sqrt((2 - 2*sqrt(1-4*x) - 3*x)/x) - 1)/2. - Benedict W. J. Irwin, Sep 24 2016

a(n) ~ 4^n/(sqrt(5*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 25 2016

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

a(n) = (1/n)*Sum_{k=0..n} binomial(n,k)*binomial(n,2*n-3*k-1). - David Nguyen, Dec 31 2016

From Alexander Burstein, Dec 12 2019: (Begin)

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

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)).

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

Maple

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

Mathematica

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

Prog

(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
(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
(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

Crossrefs

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

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

Sequence in context: A079120 A092566 A036719 * A007595 A148681 A148682

Adjacent sequences: A166132 A166133 A166134 * A166136 A166137 A166138

Keyword

nonn

Author

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

Status

approved