A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

1, 2, 4, 1, 9, 4, 22, 12, 1, 56, 35, 6, 146, 104, 24, 1, 388, 312, 86, 8, 1048, 938, 300, 40, 1, 2869, 2824, 1032, 170, 10, 7942, 8520, 3502, 680, 60, 1, 22192, 25763, 11748, 2632, 295, 12, 62510, 78064, 39072, 9926, 1330, 84, 1, 177308, 236976, 129100, 36640, 5712, 469, 14, 506008, 720574, 424344, 132960, 23660, 2352, 112, 1

Offset: 2

Author

Emeric Deutsch, Jun 02 2016

Keywords

• nonn
• tabf

Comments

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

T(n,0) = A091561(n-1).

Sum(k*T(n,k), k>=0) = A273714(n-1). This implies that the number of UHUs in all bargraphs of semiperimeter n is equal to the number of doublerises in all bargraphs of semiperimeter n-1.

Examples

			Row 4 is [4,1] 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 they have 0,1,0,0,0 UHU's.
Triangle starts
1;
2;
4,1;
9,4;
22,12,1;
56,35,6.
		

Links

• Alois P. Heinz, Rows n = 2..200, flattened
• 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. A091561, A082582, A273714.

Programs

Maple

eq := z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): 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
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0)*z^h)+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
      `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..22); # Alois P. Heinz, Jun 06 2016

Mathematica

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1, 0]*z^h] + If[t>0 || y<2, 0, b[n, y-1, -1, 0]] + If[y<1, 0, b[n-1, y, 0, If[t>0, 1, 0]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 22}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Formula

G.f.: G=G(t,z), where t marks number of UHU's and z marks semiperimeter, satisfies zG^2-(1-2z-tz^2)G+z^2 = 0.