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

2,4

Comments

The total number of descents in all bargraphs of semiperimeter n>=2. - Arnold Knopfmacher, Nov 02 2016

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.

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

Example

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.

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);

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 *)

Crossrefs

Cf. A082582, A273717.

Sequence in context: A006650 A034535 A316222 * A094806 A289596 A026639

Adjacent sequences: A273715 A273716 A273717 * A273719 A273720 A273721

Keyword

nonn

Author

Emeric Deutsch, May 29 2016

Status

approved