A273342 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having length of first column k (n>=2, k>=1).

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 13, 10, 7, 4, 1, 35, 27, 18, 11, 5, 1, 97, 75, 50, 30, 16, 6, 1, 275, 213, 143, 86, 47, 22, 7, 1, 794, 616, 416, 253, 140, 70, 29, 8, 1, 2327, 1808, 1227, 754, 424, 218, 100, 37, 9, 1, 6905, 5372, 3661, 2269, 1295, 681, 327, 138, 46, 10, 1, 20705, 16127, 11030, 6885, 3978, 2133, 1056, 475, 185, 56, 11, 1

Offset

2,4

Comments

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

Sum(k*T(n,k), k>=1) = A273343(n).

Links

Alois P. Heinz, Rows n = 2..150, 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

Formula

G.f.: G(x,z) satisfies (1 - t - tz^2 + t^2 z)G^2 - t(1 - z)(1- z - tz - tz^2)G + t^2 z^2 (1 - z) = 0 (z marks semiperimeter, x marks length of first column).

Example

Row 4 is 2,2,1 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 first-column lengths 1, 1, 2, 2, 3.

Maple

eq := (1-t-t*z^2+t^2*z)*G^2-t*(1-z)*(1-z-t*z-t*z^2)*G+t^2*z^2*(1-z) = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 23)): for n from 2 to 20 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 1 .. n-1) 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, h))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
      `if`(y<1, 0, b(n-1, y, 0, 0)*`if`(h=1, z^y, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0$2, 1)):
seq(T(n), n=2..20);  # 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, h]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, b[n - 1, y, 0, 0]*If[h == 1, z^y, 1]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][ b[n, 0, 0, 1]];
Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

Crossrefs

Cf. A082582, A273343.

Sequence in context: A184051 A121460 A105292 * A276067 A125177 A125178

Adjacent sequences: A273339 A273340 A273341 * A273343 A273344 A273345

Keyword

nonn,tabl

Author

Emeric Deutsch, May 21 2016

Status

approved