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A181307
Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (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.
0
1, 2, 6, 1, 18, 6, 54, 27, 1, 162, 108, 10, 486, 405, 64, 1, 1458, 1458, 334, 14, 4374, 5103, 1549, 117, 1, 13122, 17496, 6652, 760, 18, 39366, 59049, 27064, 4238, 186, 1, 118098, 196830, 105796, 21324, 1450, 22, 354294, 649539, 401041, 99646, 9480, 271
OFFSET
0,2
COMMENTS
Number of entries in row n is 1+floor(n/2).
Sum of entries in row n is A003480(n).
T(n,0)=2*3^{n-1}=A008776(n-1).
Sum(k*T(n,k),k>=0)=A054146(n-1).
REFERENCES
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
FORMULA
G.f. = G(t,z)=(1-z)^2/(1-4z+3z^2-tz^2).
The g.f. of column k is z^{2k}/[(1-3z)^{k+1}*(1-z)^{k-1}] (we have a Riordan array).
EXAMPLE
T(2,1)=1 because we have (1/1) (the 2-compositions are written as (top row / bottom row).
Triangle starts:
1;
2;
6,1;
18,6;
54,27,1;
162,108,10;
MAPLE
G := (1-z)^2/(1-4*z+3*z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 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 13 2010
STATUS
approved