A274491 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal segments of length 1 (n>=2, k>=0). By a horizontal segment of length 1 we mean a horizontal step that is not adjacent to any other horizontal step.

0, 1, 1, 1, 2, 1, 2, 3, 5, 4, 1, 6, 13, 9, 7, 13, 28, 32, 18, 6, 27, 68, 88, 58, 32, 2, 57, 166, 228, 207, 102, 34, 123, 394, 623, 621, 380, 166, 20, 267, 943, 1668, 1812, 1380, 630, 200, 5, 584, 2269, 4366, 5348, 4476, 2540, 967, 155

Offset

2,5

Comments

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

T(n,0) = A274493(n).

Sum(k*T(n,k), k>=0) = A274492(n).

Links

Table of n, a(n) for n=2..56.

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=G(t,z), where z marks semiperimeter and t marks horizontal segments of length 1, satisfies aG^2 + bG + c = 0, where a = z(tz-t-z), b=1-2z+z^2 - 2z^3 - 2tz^2 + 2tz^3, c=z^2(tz-t-z).

Example

Row 4 is 2,1,2 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 pictures give the values 0,2,2,0,1 for the number of horizontal segments of length 1.
Triangle starts
0,1;
1,1;
2,1,2;
3,5,4,1;
6,13,9,7;

Maple

a:=z*(t*z-t-z): b:=1-2*z+z^2-2*z^3-2*t*z^2+2*t*z^3: c:=z^2*(t*z-t-z): eq := a*G^2+b*G+c=0: g:=RootOf(eq, G): gser:=simplify(series(g, z=0, 22)): for n from 2 to 18 do P[n]:= sort(coeff(gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j=0..degree(P[n])) end do; # yields sequence in triangular form

Mathematica

nmax = 18;
a = z (t z - t - z);
b = 1 - 2z + z^2 - 2z^3 - 2t z^2 + 2t z^3;
c = z^2 (t z - t - z);
g = (-b + Sqrt[b^2 - 4 a c])/(2 a);
gser = g + O[z]^(nmax+1);
Do[Print["n = ", n]; P[n] = SeriesCoefficient[gser, {z, 0, n}], {n, 2, nmax} ];
Table[CoefficientList[P[n], t], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Jul 24 2018, from Maple *)

Crossrefs

Cf. A082582, A274492, A274493.

Sequence in context: A371335 A059346 A259439 * A076492 A127462 A106436

Adjacent sequences: A274488 A274489 A274490 * A274492 A274493 A274494

Keyword

nonn,tabf

Author

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

Status

approved