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A276067 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having length of first descent k (n>=2, 1<=k<=n-1). A descent is a maximal sequence of consecutive down steps. 1
1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 14, 9, 7, 4, 1, 41, 23, 16, 11, 5, 1, 122, 64, 39, 27, 16, 6, 1, 366, 186, 103, 66, 43, 22, 7, 1, 1105, 552, 289, 169, 109, 65, 29, 8, 1, 3356, 1657, 841, 458, 278, 174, 94, 37, 9, 1, 10251, 5013, 2498, 1299, 736, 452, 268, 131, 46, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

Number of entries in row n is n-1.

Sum of entries in row n = A082582(n).

Sum(k*T(n,k),k>=0) = A276068(n).

LINKS

Table of n, a(n) for n=2..67.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

FORMULA

G.f.:  G(t,z) = t(1-2z)(1-2z-z^2-Q)/(z(1-z-tz)), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

T(n,k)= T(n-1,k)+T(n-1,k-1) (n>=3, k>=2).

EXAMPLE

Row 4 is 2,2,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that the lengths of their first descents are 1,2,1,2,3, respectively.

Triangle starts

1;

1,1;

2,2,1;

5,4,3,1;

14,9,7,4,1.

MAPLE

G := (1/2)*t*(1-2*z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-z-t*z)): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 17 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 17 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form

MATHEMATICA

m = maxExponent = 13;

G = ((1/2) t (1 - 2z)(1 - 2z - z^2 - Sqrt[(1 - z)(1 - 3z - z^2 - z^3)])/ (z(1 - z - t z)) + O[z]^m) + O[t]^m;

Drop[CoefficientList[#/t, t]& /@ CoefficientList[G, z], 2] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

CROSSREFS

Cf. A082582, A276068.

Sequence in context: A121460 A105292 A273342 * A125177 A125178 A101975

Adjacent sequences:  A276064 A276065 A276066 * A276068 A276069 A276070

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 25 2016

STATUS

approved

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Last modified October 17 22:16 EDT 2019. Contains 328134 sequences. (Running on oeis4.)