|
|
A274490
|
|
Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).
|
|
1
|
|
|
0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 3, 1, 0, 1, 22, 8, 3, 1, 0, 1, 62, 22, 8, 3, 1, 0, 1, 178, 62, 22, 8, 3, 1, 0, 1, 519, 178, 62, 22, 8, 3, 1, 0, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,6
|
|
COMMENTS
|
Number of entries in row n is n.
Sum of entries in row n = A082582(n).
Sum_{k>=0} k*T(n,k) = A105633(n-2).
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)).
|
|
EXAMPLE
|
Row 4 is 3,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they start with 3, 1, 0, 0, 0 columns of length 1.
Triangle starts
0,1;
1,0,1;
3,1,0,1;
8,3,1,0,1;
22,8,3,1,0,1
|
|
MAPLE
|
G := (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
|
|
MATHEMATICA
|
nmax = 12;
g = (1 - 3z + z^2 + 2t z^3 - z^3 - (1-z) Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/ (2z (1 - t z));
cc = CoefficientList[g + O[z]^(nmax+1), z];
T[n_, k_] := Coefficient[cc[[n+1]], t, k];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|