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A181295
Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd entries (0<=k<=n) 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.
3
1, 0, 2, 2, 0, 5, 0, 12, 0, 12, 7, 0, 46, 0, 29, 0, 58, 0, 152, 0, 70, 24, 0, 297, 0, 466, 0, 169, 0, 256, 0, 1236, 0, 1364, 0, 408, 82, 0, 1632, 0, 4575, 0, 3870, 0, 985, 0, 1072, 0, 8160, 0, 15702, 0, 10736, 0, 2378, 280, 0, 8160, 0, 35320, 0, 51121, 0, 29282, 0, 5741, 0
OFFSET
0,3
COMMENTS
The sum of entries in row n is A003480(n).
T(2n-1,0)=0.
T(2n,0)=A003480(n).
T(n,k)=0 if n and k have opposite parities.
T(n,n)=A000129(n+1) (the Pell numbers).
Sum(k*T(n,k),k=0..n)=A181296.
For the statistics "number of even entries" see A181297.
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)^2/(1-4z^2+2z^4-2tz-t^2*z^2).
The g.f. H(t,s,z), where z marks the size of the 2-composition and t (s) marks the number of odd (even) entries, is H=1/(1-h), where h=z(t+sz)(2s+tz-sz^2)/(1-z^2)^2.
EXAMPLE
T(2,2)=5 because we have (1/1),(1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1); the 2-compositions are written as (top row / bottom row).
Triangle starts:
1;
0,2;
2,0,5;
0,12,0,12;
7,0,46,0,29;
MAPLE
G := (1-z^2)^2/(1-4*z^2+2*z^4-2*t*z-t^2*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 12 2010
STATUS
approved