A167639 - OEIS

%I #20 Oct 08 2024 22:18:57

%S 0,0,1,0,3,2,12,16,59,110,325,716,1926,4584,11887,29328,75071,188462,

%T 480778,1217876,3107689,7913082,20221903,51664040,132259190,338721180,

%U 868587021,2228677360,5723740309,14709001454,37826827606,97335031824

%N Number of peaks at even level in all Dyck paths of semilength n that have no ascents and no descents of length 1.

%C a(n) = Sum_{k>=0} k*A167637(n,k).

%H G. C. Greubel, <a href="/A167639/b167639.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: z^2/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))).

%F a(n) ~ sqrt(3/sqrt(5)-1) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - _Vaclav Kotesovec_, Mar 20 2014

%F 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

%F 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

%e 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).

%p 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);

%t 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 *)

%o (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

%Y Cf. A167636, A167637.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Nov 08 2009