cp's OEIS Frontend

A274495 The length of the longest initial sequence of the form UHUH..., summed over all bargraphs having semiperimeter n (n>=2).

%I A274495 #22 Nov 19 2024 00:42:23
%S A274495 2,3,9,23,62,171,482,1384,4036,11924,35619,107407,326521,999675,
%T A274495 3079634,9539366,29693294,92831327,291366477,917765199,2900217452,
%U A274495 9192097510,29213057684,93073003438,297215560553,951144390092,3049877146281,9797605279905
%N A274495 The length of the longest initial sequence of the form UHUH..., summed over all bargraphs having semiperimeter n (n>=2).
%H A274495 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 A274495 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 A274495 G.f.: g(z) = ((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)/(2*z*(1-z)), where Q = sqrt((1-z)*(1-3*z-z^2-z^3)).
%F A274495 a(n) = Sum_{k>=1} k*A274494(n,k).
%F A274495 D-finite with recurrence -(n+1)*(19*n-44)*a(n) +n*(43*n-65)*a(n-1) +2*(47*n^2-289*n+342)*a(n-2) +2*(-33*n^2+170*n-61)*a(n-3) +(-19*n^2+87*n+22)*a(n-4) -(33*n-31)*(n-8)*a(n-5)=0. - _R. J. Mathar_, Jul 22 2022
%e A274495 a(4) = 9 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 the sum of the lengths of their longest initial sequence of the form UHUH... is 2+4+1+1+1.
%p A274495 Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)*(1/2))/(z*(1-z)): gser := series(g, z = 0, 38): seq(coeff(gser, z, n), n = 2 .. 34);
%t A274495 terms = 28;
%t A274495 g[z_] = (((1-z)(1 - 4z^2 - 3z^3 - 2z^4) - (1 + z - z^2 - 2z^3)*Q)(1/2))/(z (1-z)) /. Q -> Sqrt[(1-z)(1 - 3z - z^2 - z^3)];
%t A274495 Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* _Jean-François Alcover_, Aug 07 2018 *)
%Y A274495 Cf. A082582, A274494.
%K A274495 nonn
%O A274495 2,1
%A A274495 _Emeric Deutsch_, _Sergi Elizalde_, Aug 26 2016