A082298 Expansion of (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).

1, 4, 20, 116, 740, 5028, 35700, 261780, 1967300, 15072836, 117297620, 924612532, 7367204260, 59240277988, 480118631220, 3917880562644, 32163325863300, 265446382860420, 2201136740855700, 18329850024033012, 153225552507991140

Offset: 0

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

Comments

More generally coefficients of (1-m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/(2*x) are given by a(0)=1 and, for n>0, a(n) = (1/n)*Sum_{k=0..n}(m+1)^k*binomial(n,k)*binomial(n,k-1).

a(n) = number of lattice paths from (0,0) to (n+1,n+1) that consist of steps (i,0) and (0,j) with i,j>=1 and that stay strictly below the diagonal line y=x except at the endpoints. (See Coker reference.) Equivalently, a(n) = number of marked Dyck (n+1)-paths where the vertices in the middle of each UU and each DD are available to be marked (or not): consider the original path as a Dyck path with a mark at each vertex where two horizontal (or two vertical) steps abut. If only the UU vertices are available for marking, then the counting sequence is the little Schroeder number A001003. - David Callan, Jun 07 2006

Hankel transform is 4^C(n+1,2). - Philippe Deléham, Feb 11 2009

a(n) is the number of Schroder paths of semilength n in which the (2,0)-steps come in 3 colors. Example: a(2)=20 because, denoting U=(1,1), H=(2,0), D=(1,-1), we have 3^2=9 paths of shape HH, 3 paths of shape HUD, 3 paths of shape UDH, 3 paths of shape UHD, and 1 path of each of the shapes UDUD, UUDD. - Emeric Deutsch, May 02 2011

(1 + 4x + 20x^2 + 116x^3 + ...) = (1 + 5x + 29x^2 + 185x^3 + ...) * 1/(1 + x + 5x^2 + 29x^3 +185x^4 + ...); where A059231 = (1, 5, 29, 185, 1257, ...) - Gary W. Adamson, Nov 17 2011

The first differences between the row sums of the triangle A226392. - J. M. Bergot, Jun 21 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
• Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
• Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
• Zhi Chen and Hao Pan, Identities involving weighted Catalan, Schroder and Motzkin paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=4, b=1.
• Curtis Coker, Enumerating a class of lattice paths, Disc. Math. (2003) Vol. 271, Issues 1-3, 13-28.
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
• John Machacek, Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats, arXiv:2105.02417 [math.CO], 2021. See Thm 4.4, G(x,F^1)
• Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See pp. 4, 22

Crossrefs

Cf. A006318, A047891, A059231, A099250.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-3*x-Sqrt(9*x^2-10*x+1))/(2*x))); // G. C. Greubel, Feb 10 2018

Maple

A082298_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od;convert(a,list)end: A082298_list(20); # Peter Luschny, May 19 2011
a := n -> `if`(n=0, 1, 4*hypergeom([1 - n, -n], [2], 4)):
seq(simplify(a(n)), n=0..20); # Peter Luschny, May 22 2017

Mathematica

gf[x_] = (1 - 3*x - Sqrt[(9*x^2 - 10*x + 1)])/(2*x); CoefficientList[Series[gf[x], {x, 0, 20}], x] (* Jean-François Alcover, Jun 01 2011 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,4^k*binomial(n,k)*binomial(n,k-1))/n)

Formula

a(0)=1, n>0 a(n) = (1/n)*Sum_{k=0..n} 4^k*binomial(n, k)*binomial(n, k-1).

a(1)=1, a(n) = 3*a(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004

a(n) = Sum_{k=0..n} 1/(n+1) Binomial(n+1,k)Binomial(2n-k,n-k)3^k. - David Callan, Jun 07 2006

From Paul Barry, Feb 01 2009: (Start)

G.f.: 1/(1-3x-x/(1-3x-x/(1-3x-x/(1-... (continued fraction);

a(n) = Sum_{k=0..n} binomial(n+k,2k)*3^(n-k)*A000108(k). (End)

a(n) = Sum_{k=0..n} A060693(n,k)*3^k. - Philippe Deléham, Feb 11 2009

D-finite with recurrence: (n+1)*a(n) = 5*(2n-1)*a(n-1)-9*(n-2)*a(n-2). - Paul Barry, Oct 22 2009

G.f.: 1/(1- 4x/(1-x/(1-4x/(1-x/(1-4x/(1-... (continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009

G.f.: (1-3*x-sqrt(9*x^2-10*x+1))/(2*x) = (1-G(0))/x; G(k) = 1+x*3-x*4/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012

a(n) ~ 3^(2*n+1)/(sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

a(n) = 4*hypergeom([1 - n, -n], [2], 4) for n>0. - Peter Luschny, May 22 2017

G.f. A(x) satisfies: A(x) = (1 + x*A(x)^2) / (1 - 3*x). - Ilya Gutkovskiy, Jun 30 2020

G.f.: (1+2*x*F(x))^2, where F(x) is the g.f. for A099250. - Alexander Burstein, May 11 2021