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 A273713 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k doublerises (n>=2, k>=0). A doublerise in a bargraph is any pair of adjacent up steps. 2
 1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 13, 9, 4, 1, 17, 32, 28, 14, 5, 1, 37, 80, 81, 50, 20, 6, 1, 82, 201, 231, 165, 80, 27, 7, 1, 185, 505, 653, 526, 295, 119, 35, 8, 1, 423, 1273, 1824, 1644, 1036, 483, 168, 44, 9, 1, 978, 3217, 5058, 5034, 3535, 1848, 742, 228, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 COMMENTS Number of entries in row n is n-1. Sum of entries in row n = A082582(n). T(n,0) = A004148(n-1) (the 2ndary structure numbers). T(n,1) = A110320(n-2). Sum(k*T(n,k), k>=0) = A273714(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 = G(t,z) satisfies zG^2 - (1 - z - tz - z^2)G + z^2 = 0. 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] and the corresponding drawings show that they have 0, 0, 1, 1, 2 doublerises. Triangle starts 1; 1,1; 2,2,1; 4,5,3,1; 8,13,9,4,1 MAPLE eq := z*G^2-(1-z-t*z-z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t),       `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+       `if`(t>0 or y<2, 0, b(n, y-1, -1))+       `if`(y<1, 0, b(n-1, y, 0))))     end: T:= n-> (p-> seq(coeff(p, z, i), i=0..n-2))(b(n, 0\$2)): seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016 MATHEMATICA b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n - 2}]][b[n, 0, 0]]; Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *) CROSSREFS Cf. A004148, A082582, A110320, A273714. Sequence in context: A202193 A105306 A183191 * A339067 A322329 A064189 Adjacent sequences:  A273710 A273711 A273712 * A273714 A273715 A273716 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, May 28 2016 STATUS approved

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Last modified April 19 02:19 EDT 2021. Contains 343105 sequences. (Running on oeis4.)