

A181338


Triangle read by rows: T(n,k) is the number of 2compositions of n having largest entry k (1<=k<=n). A 2composition 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



2, 5, 2, 12, 10, 2, 29, 41, 10, 2, 70, 152, 46, 10, 2, 169, 536, 193, 46, 10, 2, 408, 1830, 770, 198, 46, 10, 2, 985, 6120, 2972, 811, 198, 46, 10, 2, 2378, 20178, 11202, 3218, 816, 198, 46, 10, 2, 5741, 65867, 41481, 12484, 3259, 816, 198, 46, 10, 2, 13860
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OFFSET

1,1


COMMENTS

The sum of entries in row n is A003480(n).
T(n,1)=A000129(n+1) (the Pell numbers).
Sum(k*T(n,k),k=0..n)=A181339.


REFERENCES

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of Lconvex polyominoes, European Journal of Combinatorics, 28, 2007, 17241741.


LINKS

Table of n, a(n) for n=1..56.


FORMULA

G.f. for 2compositions with all entries <= k is h(k,z)=(1z)^2/(14z+2z^2+2z^{k+1}z^{2k+2}).
G.f. for 2compositions with largest entry k is f(k,z)=h(k,z)h(k1,z) (these are the column g.f.'s).
G.f. = G(t,z)=Sum(f(k,z)*t^k, k=1..infinity).


EXAMPLE

T(3,3)=2 because we have (0/3) and (3/0) (the 2compositions are written as (top row/bottom row).
Triangle starts:
2;
5,2;
12,10,2;
29,41,10,2;
70,152,46,10,2;


MAPLE

h := proc (k) options operator, arrow: (1z)^2/(14*z+2*z^2+2*z^(k+1)z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)h(k1)) end proc: G := sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 11 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form


CROSSREFS

Cf. A003480, A000129, A181339
Sequence in context: A240760 A207635 A205715 * A211175 A102469 A098886
Adjacent sequences: A181335 A181336 A181337 * A181339 A181340 A181341


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Oct 15 2010


STATUS

approved



