OFFSET
0,2
COMMENTS
Also the number of paths from (0,0) to (n,n) not rising above y=x, using steps (1,0), (0,1), (1,1) and (2,1). For example, the 7 paths to (2,2) are dd, den, end, enen, Dn, eenn and edn, where e=(1,0), n=(0,1), d=(1,1) and D=(2,1). - Brian Drake, Aug 01 2007
For another interpretation as the number of walks of a certain type, see A223092 and the link below. - N. J. A. Sloane, Mar 29 2013
Hankel transform is 3^C(n+1,2). - Paul Barry, Jan 26 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
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, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From N. J. A. Sloane, Sep 21 2012
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
N. J. A. Sloane, Illustration of initial terms of A223092 and A064641
D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations (2015) 2015:17; DOI 10.1186/s13662-014-0347-9.
FORMULA
G.f.: (1-x-sqrt(1-6x-3x^2)) / (2x(1+x)). - Brian Drake, Aug 01 2007
G.f.: 1/(1-2x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-.... (continued fraction). - Paul Barry, Jan 26 2009
a(n) = sum(i=0..n, binomial(n+i,n)*sum(j=0..n+1, binomial(j,-n+2*j-i-2)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, May 12 2011
Recurrence: (n+1)*a(n) = (5*n-4)*a(n-1) + 9*(n-1)*a(n-2) + 3*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 3*(sqrt(6)+sqrt(2))*(3+2*sqrt(3))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
G.f.: 1 / (1 - x - (x+x^2) / (1 - x - (x+x^2) / ... )) (continued fraction). - Michael Somos, Mar 30 2014
0 = a(n)*(+9*a(n+1) + 54*a(n+2) + 33*a(n+3) - 12*a(n+4)) + a(n+1)*(+78*a(n+2) + 60*a(n+3) - 27*a(n+4)) + a(n+2)*(+36*a(n+2) + 34*a(n+3) - 14*a(n+4)) + a(n+3)*(+4*a(n+3) + a(n+4)) for all n >= 0. - Michael Somos, Nov 05 2014
D-finite with recurrece (n+1)*a(n) +(-5*n+4)*a(n-1) +9*(-n+1)*a(n-2) +3*(-n+2)*a(n-3)=0. - R. J. Mathar, Oct 16 2017
EXAMPLE
The array begins
......1
....1...2
..1...5...7
1...8...22..29
G.f. = 1 + 2*x + 7*x^3 + 29*x^4 + 133*x^5 + 650*x^6 + 3319*x^7 + ...
MAPLE
A:= series( (1-x-sqrt(1-6*x-3*x^2)) / (2*x*(1+x)), x, 21): seq(coeff(A, x, i), i=0..20); # Brian Drake, Aug 01 2007
MATHEMATICA
Table[SeriesCoefficient[(1-x-Sqrt[1-6*x-3*x^2])/(2*x*(1+x)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x*(1-x)/(1+x+x^2)+O(x^(n+2))), n+1)) /* Paul Barry */
(Maxima)
a(n):=sum(binomial(n+i, n)*sum(binomial(j, -n+2*j-i-2)*binomial(n+1, j), j, 0, n+1), i, 0, n)/(n+1); /* Vladimir Kruchinin, May 12 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Floor van Lamoen, Oct 03 2001
STATUS
approved