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A026003 a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288). 9
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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

Contribution 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)

Conjecture: (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 A141630

Adjacent sequences:  A026000 A026001 A026002 * A026004 A026005 A026006

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 22 15:12 EST 2018. Contains 299454 sequences. (Running on oeis4.)