

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(i1) <= 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
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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 4paths 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



FORMULA

a(n) = Sum_{k=0..n} binomial(n1, k1)*binomial(k+1, floor((k+1)/2)).  Vladeta Jovovic, Sep 18 2003
G.f.: ((x1)^2*((1+x)/(13x))^(1/2) + x^2  1)/(2*x^2).  David Callan, Aug 16 2006
Conjecture: (n+2)*a(n) +3*(n1)*a(n1) +(n2)*a(n2) +3*(n3)*a(n3)=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(((x1)^2*((1+x)/(13x))^(1/2) + x^2  1)/(2*x^2)) \\ G. C. Greubel, May 22 2017


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



