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A273899
Triangle read by rows: for n>=1, 0<=k<=n-1, T(n,k) is the number of bargraphs of semiperimeter n having k columns of length 1.
2
0, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 4, 3, 0, 1, 13, 11, 6, 4, 0, 1, 35, 30, 18, 8, 5, 0, 1, 97, 84, 51, 26, 10, 6, 0, 1, 275, 240, 148, 76, 35, 12, 7, 0, 1, 794, 697, 435, 228, 105, 45, 14, 8, 0, 1, 2327, 2052, 1293, 688, 325, 138, 56, 16, 9, 0, 1, 6905, 6111, 3881, 2091, 1005, 440, 175, 68, 18, 10, 0, 1
OFFSET
1,7
COMMENTS
Number of entries in row n is n.
Sum of entries in row n = A082582(n).
T(n,0) = A082582(n-1).
Sum(k*T(n,k), k>=0) = A273900(n).
LINKS
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(t,z) = (1/2)(1-2z-z^2+2tz^2-2t^2z^2-2tz^3+2t^2z^3 - sqrt(1-4z+2z^2+z^4))/(1-t+t^2z+tz^2-t^2z^2), where z marks semiperimeter and t marks columns of length 1.
EXAMPLE
Row 4 is 2,2,0,1 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 pictures give the values 3,1,1,0,0 for the number of columns of length 1.
Triangle starts
0;
0,1;
1,0,1;
2,2,0,1;
5,4,3,0,1;
13,11,6,4,0,1;
35,30,18,8,5,0,1.
MAPLE
G:=((1-2*z-z^2+2*t*z^2-2*t^2*z^2-2*t*z^3+2*t^2*z^3-sqrt(1-4*z+2*z^2+z^4))*(1/2))/(1-t+t^2*z+t*z^2-t^2*z^2): Gser:= simplify(series(G, z=0, 22)): for n from 1 to 18 do P[n]:= sort(coeff(Gser, z, n)) end do: for n from 1 to 18 do seq(coeff(P[n], t, j), j=0..n-1)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>0 or y<2, 0, b(n, y-1, -1))+
`if`(y<1, 0, `if`(y=1, z, 1)*b(n-1, y, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..18); # Alois P. Heinz, Jun 07 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>0 || y<2, 0, b[n, y-1, -1]] + If[y<1, 0, If[y==1, z, 1]*b[n-1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n-1}]][ b[n, 0, 0]]; Table[T[n], {n, 1, 18}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)
CROSSREFS
Sequence in context: A267617 A158852 A188285 * A102404 A089246 A291684
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jun 07 2016
STATUS
approved