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 A026003 a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288). 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. Li-Hua Deng, Eva Y. P. Deng and Louis W. Shapiro,The Riordan Group and Symmetric Lattice Paths, arXiv:0906.1844v1 [math.CO], 2009. 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 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 *) CROSSREFS Bisections are the central Delannoy numbers A001850 and A002002 respectively. Sequence in context: A110494 A098615 A026720 * A103792 A076156 A339984 Adjacent sequences:  A026000 A026001 A026002 * A026004 A026005 A026006 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)