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%I #34 Mar 07 2023 11:29:22
%S 0,0,1,5,20,74,263,914,3134,10655,36023,121331,407610,1366926,4578365,
%T 15321750,51245820,171335458,572714527,1914159445,6397373996,
%U 21381342737,71465609723,238892470728,798659461590,2670437231049,8930385538663,29869572490093,99922049387230,334324916304050
%N The number of L-shaped corners in all bargraphs of semiperimeter n.
%C The total number of descents in all bargraphs of semiperimeter n>=2. - _Arnold Knopfmacher_, Nov 02 2016
%H G. C. Greubel, <a href="/A273718/b273718.txt">Table of n, a(n) for n = 2..500</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A. Blecher, C. Brennan and A. Knopfmacher, <a href="http://www.tandfonline.com/doi/abs/10.2989/16073606.2015.1121932">Combinatorial parameters in bargraphs</a>, Quaestiones Mathematicae, 39 (2016), 619-635.
%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 and S. Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
%F G.f.: g(z) = (1 - 4z + 3z^2 +2Q - Q)/(2zQ), where Q = sqrt(1-4z+2z^2+z^4).
%F a(n) = Sum(k*A273717(n,k), k>=0).
%F D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +2*(7*n-12)*a(n-2) +2*(-3*n+10)*a(n-3) +(n+1)*a(n-4) +3*(-n+4)*a(n-5)=0. - _R. J. Mathar_, May 30 2016
%e a(4)=1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] of which only [2,1] yields a |_ -shaped corner.
%p Q := sqrt(1-4*z+2*z^2+z^4): g := ((1-4*z+3*z^2+2*z*Q-Q)*(1/2))/(z*Q): gser := series(g, z = 0,40): seq(coeff(gser, z, n), n = 2 .. 35);
%t f[x_] := Sqrt[1 - 4*x + 2*x^2 + x^4]; CoefficientList[Series[(1 - 4*x + 3*x^2 + 2*f[x] - f[x])/(2*x*f[x]), {x, 2, 50}], x] (* _G. C. Greubel_, May 29 2016 *)
%Y Cf. A082582, A273717.
%K nonn
%O 2,4
%A _Emeric Deutsch_, May 29 2016
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