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A121483 Number of peaks at 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
1, 2, 6, 19, 56, 167, 487, 1411, 4047, 11527, 32617, 91790, 257065, 716896, 1991792, 5515535, 15227846, 41930133, 115176023, 315676425, 863475561, 2357539227, 6425887551, 17487572124, 47522431681, 128969086382, 349567320762 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = Sum(k*A121481(n,k),k=0..n).
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(1-z)(1-3z+6z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
Recurrence: (n^2 - 5*n - 20)*a(n) = (3*n^2 - 12*n - 79)*a(n-1) + (n^2 - 7*n - 16)*a(n-2) - (5*n^2 - 19*n - 138)*a(n-3) - (n^2 - 6*n - 31)*a(n-4) + (n^2 - 3*n - 24)*a(n-5). - Vaclav Kotesovec, Mar 20 2014
a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5*2^(n+2)). - Vaclav Kotesovec, Mar 20 2014
EXAMPLE
a(2)=2 because in UDUD and UUDD we have altogether 2 peaks at odd level; here U=(1,1) and D=(1,-1).
MAPLE
G:=z*(1-z)*(1-3*z+6*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*(1-x)*(1-3*x+6*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
Sequence in context: A183305 A192715 A226433 * A077834 A325918 A307564
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 02 2006
STATUS
approved

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Last modified May 30 04:46 EDT 2024. Contains 372958 sequences. (Running on oeis4.)