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A273344 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k levels. A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step. 2
1, 1, 1, 3, 2, 6, 7, 14, 19, 2, 33, 53, 11, 79, 148, 47, 1, 194, 409, 181, 10, 482, 1137, 639, 69, 1214, 3159, 2166, 360, 6, 3090, 8793, 7110, 1646, 66, 7936, 24515, 22831, 6868, 490, 2, 20544, 68443, 72145, 26893, 2918, 44, 53545, 191367, 225138, 100598, 15085, 486, 140399, 535762, 695798, 363360, 70847, 3825 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,4
COMMENTS
Sum of entries in row n = A082582(n).
LINKS
A. Blecher, C. Brennan, and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9, 2015, 297-310.
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
FORMULA
T(n,0) = A025243(n+1).
Sum(k*T(n,k), k>=1) = A273345(n).
G.f.: G(t,z) = (1-2z-z^2+2z^3-2tz^3-sqrt((1-z)(1-3z-z^2+3z^3-4tz^3+4z^4-4tz^4-4z^5+8tz^5-4t^2z^5)))/(2z(1-z+tz)); z marks semiperimeter, t marks levels. See eq. (2.4) in the Blecher et al. Ars. Math. Contemp. reference (set x = z, y = z, w = t).
EXAMPLE
Row 4 is 3,2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] having 1, 0, 0, 1, 0 levels, respectively.
Triangle starts
1;
1,1;
3,2;
6,7;
14,19,2.
MAPLE
G := (1-2*z-z^2+2*z^3-2*t*z^3-sqrt((1-z)*(1-3*z-z^2+3*z^3-4*t*z^3+4*z^4 -4*t*z^4-4*z^5+8*t*z^5-4*t^2*z^5)))/(2*z*(1-z+t*z)): Gser := simplify(series(G, z = 0, 25)): 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 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, w) option remember; expand(
`if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0))+
`if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `if`(y<1, 0,
`if`(w=1, z, 1)*b(n-1, y, 0, min(w+1, 2)))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..20); # Alois P. Heinz, Jun 04 2016
MATHEMATICA
b[n_, y_, t_, w_] := b[n, y, t, w] = Expand[If[n == 0, (1 - t), If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, If[w == 1, z, 1]*b[n - 1, y, 0, Min[w + 1, 2]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)
CROSSREFS
Sequence in context: A269852 A329691 A293204 * A370665 A127717 A210236
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 21 2016
STATUS
approved

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Last modified July 20 17:32 EDT 2024. Contains 374459 sequences. (Running on oeis4.)