A273350 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k up steps (n >= 2, k >= 1).

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 16, 7, 1, 1, 15, 40, 31, 9, 1, 1, 21, 85, 105, 51, 11, 1, 1, 28, 161, 295, 219, 76, 13, 1, 1, 36, 280, 721, 771, 396, 106, 15, 1, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 19, 1

Offset: 2

Author

Emeric Deutsch, Jun 02 2016

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n-1.

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

Sum_{k>=1} k*T(n,k) = A273351(n).

Connection with A145904 should be explored.

This seems to be a duplicate of A145904. - Alois P. Heinz, Jun 04 2016. [This could probably be proved by showing that the g.f.s are the same. - N. J. A. Sloane, Jul 02 2016]

Examples

			Row 4 is 1,3,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 1,2,2,2,3 up steps.
Triangle starts
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 16,  7,  1;
  1, 15, 40, 31,  9,  1.
		

Links

• A. Blecher, C. Brennan and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, 39 (2016), 619-635.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A082582, A145904, A273351.

Programs

Maple

eq := z*G^2-(1-z-y*z-y*z^2)*G+y*z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], y, j), j = 1 .. n-1) end do; # yields sequence in triangular form

Mathematica

G = (-1 + z + y z + y z^2 + Sqrt[-4y z^3 + (-1 + z + y z + y z^2)^2])/(2z);
Rest @ CoefficientList[#, y]& /@ Drop[-CoefficientList[G + O[z]^13, z], 2] // Flatten (* Jean-François Alcover, Oct 24 2019 *)

Maxima

T(n,m):=if n<=m then 0 else 1/(n-m)*sum(binomial(n-m,i-1)*binomial(n-m,i)*binomial(n-i-1,m-i),i,0,m); /* Vladimir Kruchinin, Jan 15 2018 */

Formula

G.f.: G=G(t,z) satisfies zG^2-(1-z-yz-yz^2)G+yz^2=0 (z marks semiperimeter, y marks up-steps).

T(n,m) = (1/(n-m))*Sum_{i=0..m} C(n-m,i-1)*C(n-m,i)*C(n-i-1,m-i). - Vladimir Kruchinin, Jan 15 2018