A274490 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 3, 1, 0, 1, 22, 8, 3, 1, 0, 1, 62, 22, 8, 3, 1, 0, 1, 178, 62, 22, 8, 3, 1, 0, 1, 519, 178, 62, 22, 8, 3, 1, 0, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1

Offset: 2

Author

Emeric Deutsch, Jun 25 2016

Keywords

• nonn
• tabf

Comments

Number of entries in row n is n.

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

T(n,0) = A188464(n-3) (n>=3).

Sum_{k>=0} k*T(n,k) = A105633(n-2).

Examples

			Row 4 is 3,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they start with 3, 1, 0, 0, 0 columns of length 1.
Triangle starts
0,1;
1,0,1;
3,1,0,1;
8,3,1,0,1;
22,8,3,1,0,1
		

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, A105633, A188464.

Programs

Maple

G := (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)): 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, k), k = 0 .. n-1) end do; # yields sequence in triangular form

Mathematica

nmax = 12;
g = (1 - 3z + z^2 + 2t z^3 - z^3 - (1-z) Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/ (2z (1 - t z));
cc = CoefficientList[g + O[z]^(nmax+1), z];
T[n_, k_] := Coefficient[cc[[n+1]], t, k];
Table[T[n, k], {n, 2, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

Formula

G.f.: (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)).