OFFSET
0,5
COMMENTS
The terms a(n), starting from n=3, are the partial sums of the sequence A089735.
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 05 2011
STATUS
approved