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%I #42 Feb 10 2022 09:33:04
%S 1,1,3,5,13,25,63,129,321,681,1683,3653,8989,19825,48639,108545,
%T 265729,598417,1462563,3317445,8097453,18474633,45046719,103274625,
%U 251595969,579168825,1409933619,3256957317,7923848253,18359266785,44642381823
%N a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288).
%C 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
%C 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
%C 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
%D 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
%D Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
%H Vincenzo Librandi, <a href="/A026003/b026003.txt">Table of n, a(n) for n = 0..200</a>
%H Axel Bacher, <a href="https://arxiv.org/abs/1802.06030">Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths</a>, arXiv:1802.06030 [cs.DS], 2018.
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry1/barry411.html">The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
%H Li-Hua Deng, Eva Y. P. Deng and Louis W. Shapiro,<a href="https://arxiv.org/abs/0906.1844">The Riordan Group and Symmetric Lattice Paths</a>, arXiv:0906.1844v1 [math.CO], 2009.
%F G.f.: (sqrt((x^2-2*x-1)/(x^2+2*x-1))-1)/2/x. - _Vladeta Jovovic_, Apr 27 2003
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(n-2k, floor((n-2k)/2)). - _Paul Barry_, Jul 30 2005
%F From _Paul Barry_, Mar 01 2010: (Start)
%F G.f.: 1/(1-x-2x^2/(1-x^2/(1-2x^2/(1-x^2/(1-2x^2/(1-... (continued fraction),
%F 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)
%F 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
%F a(n) ~ (1+sqrt(2))^(n+1) / (2^(3/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 13 2014
%p 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
%t 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 *)
%Y Bisections are the central Delannoy numbers A001850 and A002002 respectively.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_
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