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A273718
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The number of L-shaped corners in all bargraphs of semiperimeter n.
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2
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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
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OFFSET
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2,4
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COMMENTS
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The total number of descents in all bargraphs of semiperimeter n>=2. - Arnold Knopfmacher, Nov 02 2016
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 2..500
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, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016
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FORMULA
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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).
Conjecture: (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
Conjecture: -(n+1)*(4*n-15)*a(n) +2*(8*n^2-28*n+15)*a(n-1) -2*(4*n-9)*(n-3)*a(n-2) +4*(n-3)*a(n-3) -(4*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, May 30 2016
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EXAMPLE
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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.
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MAPLE
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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
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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 *)
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CROSSREFS
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Cf. A082582, A273717.
Sequence in context: A006650 A034535 A316222 * A094806 A289596 A026639
Adjacent sequences: A273715 A273716 A273717 * A273719 A273720 A273721
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, May 29 2016
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STATUS
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approved
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