A104632 -id:A104632 - cp's OEIS frontend

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, 31921622746680, 120970706601255

Offset: 1

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

nonn

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

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.

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

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

[(&+[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

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

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

vector(30, n, sum(k=0,n, binomial(n,k)*binomial(n,2*n-3*k-1))/n ) \\ G. C. Greubel, Dec 12 2019

[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

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: (Start)
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)

A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

Original entry on oeis.org

0, 1, 4, 18, 80, 365, 1686, 7875, 37080, 175725, 837100, 4004770, 19227924, 92599533, 447118140, 2163837030, 10492874384, 50972030189, 248000853348, 1208335275170, 5894873067200, 28791371852145, 140768761906190

Offset: 0

T. D. Noe, Mar 17 2005

In the triangle of pentanomial coefficients, these numbers are in the column next to the central pentanomial coefficients, A005191. Note that for n>0, n divides a(n). This divisibility property is also true for the triangle of trinomial coefficients, A027907, but apparently for no other triangle of m-nomial coefficients. The quotient a(n)/n is in A104632.

			G.f. = x + 4*x^2 + 18*x^3 + 80*x^4 + 365*x^5 + 1686*x^6 + 7875*x^7 + ... - _Michael Somos_, Aug 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Andrei G. Pronko, Periodic Motzkin chain: Ground states and symmetries, arXiv:2504.00835 [math-ph], 2025. See p. 16.

Cf. A035343 (triangle of pentanomial coefficients).

Cf. A104632.

P:=PolynomialRing(Integers()); [n eq 0 select 0 else Coefficients((1+x+x^2+x^3+x^4)^n)[2*n+2]: n in [0..22]]; // Bruno Berselli, Nov 17 2011

f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1], {n, 30}]

x='x+O('x^30); concat([0], Vec(sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))))) \\ G. C. Greubel, Aug 12 2018

G.f.: sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))). - Mark van Hoeij, Nov 16 2011
From Vaclav Kotesovec, Oct 17 2012: (Start)
Recurrence: 2*(n-1)*(2*n+1)*a(n) = (19*n^2 - 19*n + 2)*a(n-1) + 5*(2*n^2 - 3*n - 1)*a(n-2) - 25*(n-2)*n*a(n-3).
a(n) ~ 5^n/(2*sqrt(Pi*n)). (End)
a(n) = n * A104632(n) for n>=0. - Michael Somos, Aug 12 2018

cp's OEIS Frontend

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.

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