A026003 a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288).

1, 1, 3, 5, 13, 25, 63, 129, 321, 681, 1683, 3653, 8989, 19825, 48639, 108545, 265729, 598417, 1462563, 3317445, 8097453, 18474633, 45046719, 103274625, 251595969, 579168825, 1409933619, 3256957317, 7923848253, 18359266785, 44642381823

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Number of lattice paths from (0,0) to the line x=n consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis (i.e. left factors of Schroeder paths); for example, a(3)=5, counting the paths UUU,UUD,UDU,HU and UH. - Emeric Deutsch, Oct 27 2002

Transform of A001405 by |A049310(n,k)|, that is, transform of central binomial coefficients C(n,floor(n/2)) by Chebyshev mapping which takes a sequence with g.f. g(x) to the sequence with g.f. (1/(1-x^2))g(x/(1-x^2)). - Paul Barry, Jul 30 2005

The Kn1p sums, p >= 1, see A180662, of the Schroeder triangle A033877 (offset 0) are all related to A026003, e.g. Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - (2*n+7), Kn14(n) = A026003(n+6) - (2*n^2+18*n+41), Kn15(n) = A026003(n+8) - (4*n^3+66*n^2+368*n+693)/3, etc.. - Johannes W. Meijer, Jul 15 2013

References

• L. Ericksen, Lattice path combinatorics for multiple product identities, J. Stat. Plan. Infer. 140 (2010) 2213-2226 doi:10.1016/j.jspi.2010.01.017
• Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.
• Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
• Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 20, 29.
• Li-Hua Deng, Eva Y. P. Deng and Louis W. Shapiro,The Riordan Group and Symmetric Lattice Paths, arXiv:0906.1844v1 [math.CO], 2009.

Crossrefs

Bisections are the central Delannoy numbers A001850 and A002002 respectively.

Programs

Maple

A026003 :=n -> add(binomial(n-k, k) * binomial(n-2*k, floor((n-2*k)/2)), k=0..floor(n/2)): seq(A026003(n), n=0..30); # Johannes W. Meijer, Jul 15 2013

Mathematica

CoefficientList[Series[(Sqrt[(x^2-2*x-1)/(x^2+2*x-1)]-1)/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

G.f.: (sqrt((x^2-2*x-1)/(x^2+2*x-1))-1)/2/x. - Vladeta Jovovic, Apr 27 2003

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(n-2k, floor((n-2k)/2)). - Paul Barry, Jul 30 2005

From Paul Barry, Mar 01 2010: (Start)

G.f.: 1/(1-x-2x^2/(1-x^2/(1-2x^2/(1-x^2/(1-2x^2/(1-... (continued fraction),

G.f.: 1/(1-x-x^2-x^2/(1-x^2-x^2/(1-x^2-x^2/(1-x^2-x^2/(1-... (continued fraction). (End)

D-finite with recurrence (n+1)*a(n) -2*a(n-1) +6*(-n+1)*a(n-2) -2*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Nov 30 2012

a(n) ~ (1+sqrt(2))^(n+1) / (2^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014