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

Author

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

Keywords

• nonn
• tabf

Comments

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

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

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

Examples

			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;
		

Links

• 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

Crossrefs

Cf. A082582, A274492, A274493.

Programs

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 *)

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).