A181330 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's in the top row 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.
1, 1, 1, 3, 3, 1, 8, 10, 5, 1, 21, 32, 21, 7, 1, 55, 99, 80, 36, 9, 1, 144, 299, 286, 160, 55, 11, 1, 377, 887, 978, 650, 280, 78, 13, 1, 987, 2595, 3236, 2482, 1275, 448, 105, 15, 1, 2584, 7508, 10438, 9054, 5377, 2261, 672, 136, 17, 1, 6765, 21526, 32991, 31882
Offset: 0
Examples
T(2,1)=3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)). Triangle starts: 1; 1,1; 3,3,1; 8,10,5,1; 21,32,21,7,1; 55,99,80,36,9,1;
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.
Programs
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Maple
G := (1-z)^2/(1-3*z+z^2-t*z*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
Formula
G.f.: G(t,x) = (1-x)^2/(1-3*x+x^2-t*x(1-x)).
The g.f. of column k is x^k*(1-x)^(k+2)/(1-3*x+x^2)^(k+1) (we have a Riordan array).
T(n,k) = 3*T(n-1,k) +T(n-1,k-1) -T(n-2,k) -T(n-2,k-1), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 26 2013
Comments