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, 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

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