A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=n).

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

Offset: 1

Author

Emeric Deutsch, Oct 15 2010

Keywords

• nonn
• tabl

Comments

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.

Examples

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

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.

Crossrefs

Cf. A003480, A000129, A181339

Programs

Maple

h := proc (k) options operator, arrow: (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)-h(k-1)) 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

Formula

G.f. for 2-compositions with all entries <= k is h(k,z) = (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)).

G.f. for 2-compositions with largest entry k is f(k,z) = h(k,z)-h(k-1,z) (these are the column g.f.'s).

G.f.: G(t,z) = Sum_{k>=1} f(k,z)*t^k.

Sum_{k=0..n} T(n,k) = A003480(n).

T(n,1) = A000129(n+1) (the Pell numbers).

Sum_{k=0..n} k*T(n,k) = A181339(n).