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A121530
Number of double rises at an odd 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.
2
0, 1, 4, 14, 47, 148, 454, 1359, 4004, 11644, 33521, 95696, 271300, 764605, 2143964, 5985186, 16643779, 46124692, 127433562, 351106955, 964976460, 2646158176, 7241414949, 19779499584, 53933402472, 146828245753, 399137621524
OFFSET
1,3
COMMENTS
a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).
LINKS
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
FORMULA
G.f.=z^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
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
EXAMPLE
a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).
MAPLE
g:=z^2*(1-2*z-z^2+4*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=1..30);
MATHEMATICA
Rest[CoefficientList[Series[x^2*(1-2*x-x^2+4*x^3-3*x^4)/(1+x)/(1-3*x+x^2)^2 /(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 05 2006
STATUS
approved