

A166135


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


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 2nomial trees, Motzkin integers on the edge of cut 3nomial trees.
a(n) is the number of increasing unarybinary trees with associated permutation that avoids 213. For more information about increasing unarybinary 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 xaxis.  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, Michael Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.
Jérémie Bettinelli, Éric Fusy, Cécile Mailler, Lucas Randazzo, A bijective study of Basketball walks, arXiv:1611.01478 [math.CO], 2016.
Rick Jarosh, Illustration of 4nomial 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,nk)*C(2*k2,k1).  Vladimir Kruchinin, May 12 2012
G.f.: (sqrt((2  2*sqrt(14*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^253*n+24)*a(n1) + 6*(13*n^2+43*n36)*a(n2)  108*(2*n5)*(n3)*a(n3) = 0.  R. J. Mathar, Oct 08 2016
a(n) = (1/n)*Sum_{k=0..n} binomial(n,k)*binomial(n,2*n3*k1).  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 pseudoinvolution, i.e., the series reversion of x*(1 + x*A(x)) is x*(1  x*A(x)).
The Bsequence 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*n3*k1), k=0..n)/n, n=1..30); # G. C. Greubel, Dec 12 2019


MATHEMATICA

Rest[CoefficientList[Series[(Sqrt[(22Sqrt[14x]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*n3*k1))/n ) \\ G. C. Greubel, Dec 12 2019
(MAGMA) [(&+[Binomial(n, k)*Binomial(n, 2*n3*k1): k in [0..n]])/n : n in [1..30]]; // G. C. Greubel, Dec 12 2019
(Sage) [sum(binomial(n, k)*binomial(n, 2*n3*k1) 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, 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



