cp's OEIS Frontend

A166287 Number of peak plateaux in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)).

%I A166287 #12 Mar 23 2017 04:36:59
%S A166287 0,0,0,1,3,8,21,53,133,334,839,2112,5329,13475,34143,86674,220400,
%T A166287 561309,1431522,3655480,9345287,23916622,61267207,157088278,403103955,
%U A166287 1035192885,2660312103,6841157380,17603254230,45321606641,116748360064
%N A166287 Number of peak plateaux in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)).
%C A166287 A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.
%H A166287 G. C. Greubel, <a href="/A166287/b166287.txt">Table of n, a(n) for n = 0..1000</a>
%F A166287 a(n) = Sum_{k>=0} k*A166285(n,k).
%F A166287 G.f.: G=(1-z-z^2-h)/[2(1-z)h], where h = sqrt((1-3z+z^2)(1+z+z^2)).
%F A166287 a(n) ~ (3+sqrt(5))^n / (5^(1/4) * sqrt(Pi*n) * 2^(n+1)). - _Vaclav Kotesovec_, Mar 20 2014
%F A166287 Conjecture: n*a(n) +(-4*n+3)*a(n-1) +3*(n-1)*a(n-2) +(n-9)*a(n-3) +(3*n-5)*a(n-4) +(-3*n+7)*a(n-5) +(-2*n+13)*a(n-6) +(n-6)*a(n-7)=0. - _R. J. Mathar_, Jun 14 2016
%F A166287 Conjecture: n*(2*n-5)*(2*n-7)*a(n) -(2*n-7)*(6*n^2-17*n+8)*a(n-1) +(n-2)*(4*n^2-16*n-1)*a(n-2) +(-4*n^3+32*n^2-71*n+44)*a(n-3) +(2*n-3) (6*n^2-37*n+54)*a(n-4) -(n-4)*(2*n-3)*(2*n-5)*a(n-5)=0. - _R. J. Mathar_, Jun 14 2016
%e A166287 a(4)=3 because we have UDUDUDUD, UDUDUUDD, UDUUDDUD, UD(UUDUDD), UUDDUDUD, UUDDUUDD, (UUDUDD)UD, (UUDUDUDD) (the 3 peak plateaux are shown between parentheses).
%p A166287 h := sqrt((1-3*z+z^2)*(1+z+z^2)): G := ((1-z-z^2-h)*1/2)/((1-z)*h): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%t A166287 CoefficientList[Series[((1-x-x^2-Sqrt[(1-3*x+x^2)*(1+x+x^2)])*1/2)/((1-x)*Sqrt[(1-3*x+x^2)*(1+x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A166287 (PARI) x='x+O('x^50); concat([0,0,0], Vec(((1-x-x^2-sqrt((1-3*x+x^2)*(1+x+x^2))))/(2*(1-x)*sqrt((1-3*x+x^2)*(1+x+x^2))))) \\ _G. C. Greubel_, Mar 22 2017
%Y A166287 Cf. A166285.
%K A166287 nonn
%O A166287 0,5
%A A166287 _Emeric Deutsch_, Oct 12 2009