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A082298
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G.f.: (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).
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10
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1, 4, 20, 116, 740, 5028, 35700, 261780, 1967300, 15072836, 117297620, 924612532, 7367204260, 59240277988, 480118631220, 3917880562644, 32163325863300, 265446382860420, 2201136740855700, 18329850024033012, 153225552507991140
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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.
P. Barry, A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations , J. Int. Seq. 14 (2011) # 11.3.8.
Z. Chen, H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=4, b=1.
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 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.
J. Machacek, Lattice walks ending on a coordinate hyperplane avoiding..., arXiv:2105.02417 Thm 4.4, G(x,F^1)
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<1, 1, sum(k=0, n, 4^k*binomial(n, k)*binomial(n, k-1))/n)
(MAGMA) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-3*x-Sqrt(9*x^2-10*x+1))/(2*x))) // G. C. Greubel, Feb 10 2018
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CROSSREFS
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Cf. A006318, A047891, A059231, A099250.
Sequence in context: A120915 A165311 A100328 * A129378 A078944 A158900
Adjacent sequences: A082295 A082296 A082297 * A082299 A082300 A082301
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre, May 10 2003
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STATUS
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approved
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