login
A181299
Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
2
1, 2, 6, 1, 20, 4, 64, 17, 1, 206, 68, 6, 662, 261, 32, 1, 2128, 976, 152, 8, 6840, 3577, 675, 51, 1, 21986, 12912, 2860, 280, 10, 70670, 46049, 11704, 1406, 74, 1, 227156, 162628, 46632, 6632, 460, 12, 730152, 569705, 181877, 29866, 2570, 101, 1, 2346942
OFFSET
0,2
COMMENTS
Row n contains 1+floor(n/2) entries.
The sum of entries in row n is A003480(n).
T(n,0) = A181301(n).
Sum_{k>=0} k*T(n,k) = A181300(n).
LINKS
G. Castiglione, A. Frosini, E. Munarini, A. Restivo, and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
FORMULA
G.f.: G(t,z) = (1+z)*(1-z)^2/(1-3*z-z^2+z^3-t*(1-z)*z^2).
EXAMPLE
T(3,1)=4 because we have (1,1/1,0),(1,0/1,1),(1,1/0,1),(0,1/1,1) (the 2-compositions are written as (top row/bottom row).
Triangle starts:
1;
2;
6,1;
20,4;
64,17,1;
MAPLE
G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 12 2010
STATUS
approved