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 A026135 Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120. 4
 1, 2, 5, 14, 39, 110, 312, 890, 2550, 7334, 21161, 61226, 177575, 516114, 1502867, 4383462, 12804429, 37452870, 109682319, 321563658, 943701141, 2772060618, 8149661730, 23978203662, 70600640796, 208014215066, 613266903927 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the total number of rows of consecutive peaks in all Motzkin (n+2)-paths. For example, with U=upstep, D=downstep, F=flatstep, the path FU(UD)FU(UDUDUD)DD(UD) contains 3 rows of peaks (in parentheses). The 9 Motzkin 4-paths are FFFF, FF(UD), F(UD)F, FUFD, (UD)FF, (UDUD), UFDF, UFFD, U(UD)D, containing a total of 5 rows of peaks and so a(2)=5. - David Callan, Aug 16 2006 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - Vladeta Jovovic, Sep 18 2003 G.f.: ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - David Callan, Aug 16 2006 G.f. = (1+z)*(1+z^2)/(1-z) where z=x*A001006(x). [From R. J. Mathar, Jul 07 2009] Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n-2)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013 MATHEMATICA CoefficientList[Series[((x - 1)^2*((1 + x)/(1 - 3 x))^(1/2) + x^2 - 1)/(2*x^2), {x, 0, 50}], x] (* G. C. Greubel, May 22 2017 *) PROG (PARI) x='x+O('x^50); Vec(((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2)) \\ G. C. Greubel, May 22 2017 CROSSREFS First differences are in A025566, second differences in A005773. Pairwise sums of A025179. Sequence in context: A202207 A132834 A000641 * A201778 A367655 A105641 Adjacent sequences: A026132 A026133 A026134 * A026136 A026137 A026138 KEYWORD nonn AUTHOR Clark Kimberling EXTENSIONS More terms from David Callan, Aug 16 2006 Typo in a(19) corrected by R. J. Mathar, Jul 07 2009 STATUS approved

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Last modified December 7 02:45 EST 2023. Contains 367622 sequences. (Running on oeis4.)