A273718 The number of L-shaped corners in all bargraphs of semiperimeter n.
0, 0, 1, 5, 20, 74, 263, 914, 3134, 10655, 36023, 121331, 407610, 1366926, 4578365, 15321750, 51245820, 171335458, 572714527, 1914159445, 6397373996, 21381342737, 71465609723, 238892470728, 798659461590, 2670437231049, 8930385538663, 29869572490093, 99922049387230, 334324916304050
Offset: 2
Keywords
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 2..500
- Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
- A. Blecher, C. Brennan and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, 39 (2016), 619-635.
- M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. Appl. Math., 31, 2003, 86-112.
- Emeric Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
Programs
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Maple
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);
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Mathematica
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 *)
Formula
G.f.: g(z) = (1 - 4z + 3z^2 +2Q - Q)/(2zQ), where Q = sqrt(1-4z+2z^2+z^4).
a(n) = Sum(k*A273717(n,k), k>=0).
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
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