A273722 - OEIS

%I #20 Aug 07 2018 09:35:45

%S 0,0,0,0,1,7,34,143,558,2083,7559,26913,94547,328943,1136218,3903245,

%T 13352270,45524764,154811018,525345268,1779722313,6020903806,

%U 20346143381,68691126090,231732871764,781268589267,2632605033729,8867115559325,29855369535397

%N The number of valleys of width 1 (i.e., DHU configurations, where U=(0,1), H(1,0), D=(0,-1)) in all bargraphs of semiperimeter n (n>=2).

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016

%F G.f.: g(z)=(1-5z+6z^2-z^3+z^4-(1-3z+z^2)Q)/(2zQ), where Q = sqrt(1-4z+2z^2+z^4).

%F a(n) = Sum_{k >= 1} k*A273721(n,k).

%F Conjecture: -(n-6) *(2*n-7) *(2*n-9) *(n+1)*a(n) +2*(n-3) *(2*n-9) *(4*n^2-24*n+21)*a(n-1) +2*(-4*n^4+56*n^3-289*n^2+651*n-547) *a(n-2) +4*(2*n-5) *(n-4)*a(n-3) -(n-4) *(n-5) *(2*n-5) *(2*n-7) *a(n-4)=0. - _R. J. Mathar_, Jun 02 2016

%e a(4)=0 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]  and the corresponding drawings show that they have no 1-width valleys.

%e a(6)=1 because there is only one bargraph of semiperimeter 6 having a 1-width valley (it corresponds to the composition [2,1,2]).

%p Q:=sqrt(1-4*z+2*z^2+z^4): g:=((1-5*z+6*z^2-z^3+z^4-(1-3*z+z^2)*Q)*(1/2))/(z*Q): gser:= series(g,z=0,40): seq(coeff(gser, z, n), n = 2 .. 35);

%t terms = 29;

%t g[z_] = (1 - 5z + 6z^2 - z^3 + z^4 - (1 - 3z + z^2) Q)/(2z Q) /. Q -> Sqrt[1 - 4z + 2z^2 + z^4];

%t Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* _Jean-François Alcover_, Aug 07 2018 *)

%Y Cf. A082582, A273721.

%K nonn

%O 2,6

%A _Emeric Deutsch_, Jun 01 2016