A121532 Number of double rises at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
0, 0, 1, 6, 24, 87, 290, 926, 2861, 8640, 25634, 75015, 217100, 622620, 1772097, 5011394, 14093980, 39448623, 109954398, 305344314, 845165725, 2332485420, 6420202246, 17629525871, 48304680504, 132092031672, 360557665825
Offset: 1
Keywords
Examples
a(3)=1 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UU/UDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).
Links
- E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
- Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1).
Programs
-
Magma
R
:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) )); // G. C. Greubel, May 24 2019 -
Maple
g:=z^3*(1-3*z^2+2*z^3-z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,35): seq(coeff(gser,z,n),n=1..32);
-
Mathematica
Rest[CoefficientList[Series[x^3*(1-3*x^2+2*x^3-x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 20 2014 *) -
PARI
my(x='x+O('x^30)); concat([0,0], Vec(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)))) \\ G. C. Greubel, May 24 2019 -
Sage
a=(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) ).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 24 2019
Formula
a(n) = Sum_{k>=0} k*A121531(n,k).
G.f.: x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)). [Corrected by Georg Fischer, May 24 2019]
a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021