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 (callan(AT)stat.wisc.edu), Aug 16 2006

G.f. ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006

nonn,new

nonn

More terms from David Callan (callan(AT)stat.wisc.edu), Aug 16 2006

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 18 2003

nonn,new

nonn

Pairwise sums of A025179.

nonn,new

nonn

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*binomial(k+1,floor((k+1)/2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 18 2003

nonn,new

nonn

First differences are in A025566, second differences in A005773.

nonn,new

nonn

nonn,new

nonn

Clark Kimberling (ck6@cedar.(AT)evansville.edu)

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.

1, 2, 5, 14, 39, 110, 312, 890, 2550, 7334, 21161, 61226, 177575, 516114, 1502867, 4383462, 12804429, 37452870, 109682319, 321564658, 943701141, 2772060618, 8149661730, 23978203662, 70600640796, 208014215066, 613266903927

0,2

nonn

Clark Kimberling (ck6@cedar.evansville.edu)

approved