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A167639
Number of peaks at even level in all Dyck paths of semilength n that have no ascents and no descents of length 1.
3
0, 0, 1, 0, 3, 2, 12, 16, 59, 110, 325, 716, 1926, 4584, 11887, 29328, 75071, 188462, 480778, 1217876, 3107689, 7913082, 20221903, 51664040, 132259190, 338721180, 868587021, 2228677360, 5723740309, 14709001454, 37826827606, 97335031824
OFFSET
0,5
COMMENTS
a(n) = Sum_{k>=0} k*A167637(n,k).
LINKS
FORMULA
G.f.: z^2/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))).
a(n) ~ sqrt(3/sqrt(5)-1) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n - 1) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
D-finite with recurrence (-n+2)*a(n) +(n-3)*a(n-1) +2*(2*n-5)*a(n-2) +(n-5)*a(n-3) +(-3*n+11)*a(n-5) +(n-4)*a(n-6)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(5)=2 because U(UD)DUUUDDD, UUUDDDU(UD)D, UUUDDUUDDD, and UUUUUDDDDD have 1 + 1 + 0 + 0 = 2 even-level peaks (shown between parentheses).
MAPLE
G := z^2/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
MATHEMATICA
CoefficientList[Series[x^2/((1+x-x^2)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0], Vec(x^2/((1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2))))) \\ G. C. Greubel, Feb 12 2017
CROSSREFS
Sequence in context: A253246 A152550 A114798 * A113205 A136657 A006774
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 08 2009
STATUS
approved