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%I #27 Jul 22 2022 10:26:37
%S 1,1,3,8,23,69,212,662,2091,6661,21359,68850,222892,724175,2360010,
%T 7711148,25252819,82863807,272385447,896774552,2956599075,9760032991,
%U 32255829642,106713308118,353381245728,1171248042277,3885122245389,12896869926038
%N The number of peaks of width 1 (i.e., UHD 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. 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^2*(1 + z^2 + Q)/(2*Q), where Q = sqrt(1-4*z+2*z^2+z^4).
%F a(n) = Sum(k*A273715(n,k), k>=1).
%F D-finite with recurrence (n-2)*(2*n-9)*a(n) +(2*n^2-29*n+75)*a(n-1) -6*(2*n-7)*(3*n-17)*a(n-2) +10*(2*n-9)*(n-5)*a(n-3) +(2*n-5)*(n-6)*a(n-4) +5*(2*n-7)*(n-7)*a(n-5)=0. - _R. J. Mathar_, Jun 02 2016
%e a(4)=3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they have 0, 1, 1, 0, 1 peaks of width 1..
%p g := (1/2)*z^2*(1+z^2+sqrt(1-4*z+2*z^2+z^4))/sqrt(1-4*z+2*z^2+z^4): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 2 .. 35);
%t b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[t > 0, 1, 0]]]]];
%t a[n_] := Module[{cc}, cc = Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; cc.Range[0, Length[cc]-1]];
%t Table[a[n], {n, 2, 29}] (* _Jean-François Alcover_, Jul 25 2018, after A273715 and _Alois P. Heinz_ *)
%Y Cf. A082582, A273715.
%K nonn
%O 2,3
%A _Emeric Deutsch_, May 28 2016
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