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A187260 Number of uh^jd's for some j>0, starting at level 0, where u=(1,1), h=(1,0), and d=(1,-1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology). 1
0, 0, 0, 1, 3, 6, 12, 25, 53, 115, 255, 575, 1315, 3043, 7111, 16756, 39766, 94961, 228003, 550081, 1332839, 3241930, 7913028, 19375635, 47579847, 117149125, 289142441, 715253644, 1773011502, 4403539181, 10956537307, 27307002454, 68164324150, 170404155586, 426584025250, 1069289177950 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The terms a(n), starting from n=3, are the partial sums of the sequence A089735.
LINKS
FORMULA
G.f.: z^3*g^2/(1-z), where g=1+z*g+z^2*g*(g-1).
a(n) = Sum_{k>=0} k*A098071(n,k).
From Vaclav Kotesovec, May 29 2022: (Start)
G.f.: (-1 + x - x^2 + sqrt((1 + (-3 + x)*x) * (1 + x + x^2)))^2 / (4*(1-x)*x).
a(n) ~ 5^(1/4) * phi^(2*n-1) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence (n+1)*a(n) +(-4*n+1)*a(n-1) +(5*n-8)*a(n-2) +(-5*n+18)*a(n-3) +(5*n-22)*a(n-4) +(-5*n+32)*a(n-5) +(4*n-31)*a(n-6) +(-n+9)*a(n-7)=0. - R. J. Mathar, Jul 22 2022
EXAMPLE
a(4)=3 because the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely hhhh, h(uhd), (uhd)h, and (uhhd) contain 0+1+1+1 subwords of type uh^ju for some j>0, starting at level 0 (shown between parentheses.
MAPLE
eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): F := z^3*g^2/(1-z): Fser := series(F, z = 0, 38): seq(coeff(Fser, z, n), n = 0 .. 35);
MATHEMATICA
CoefficientList[Series[(-1 + x - x^2 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)])^2 / (4*(1 - x)*x), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)
CROSSREFS
Sequence in context: A293313 A267539 A243722 * A243723 A243724 A243725
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 05 2011
STATUS
approved

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Last modified February 25 16:57 EST 2024. Contains 370332 sequences. (Running on oeis4.)