

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<=n1). 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 n1.
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. BousquetMélou and A. Rechnitzer, The siteperimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86112.
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(12z)(12zz^2Q)/(z(1ztz)), where Q = sqrt((1z)(13zz^2z^3)).
T(n,k)= T(n1,k)+T(n1,k1) (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*(12*z)*(12*zz^2sqrt((1z)*(13*zz^2z^3)))/(z*(1zt*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 .. n1) 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 (* JeanFrançois 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



