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

0,4

Comments

The sum of entries in row n is A003480(n).

T(n,0) = A000045(2n) (n>=1), Fibonacci numbers.

T(n,1) = A038731(n-1) (n>=1).

Sum(k*T(n,k), k>=0) = A181331.

For the statistic "number of nonzero entries in the top row" see A181332.

Links

Table of n, a(n) for n=0..58.

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.

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

Example

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;

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

Crossrefs

Cf. A003480, A000045, A038731, A181331, A181332.

Sequence in context: A160332 A293294 A200342 * A350253 A262143 A284554

Adjacent sequences: A181327 A181328 A181329 * A181331 A181332 A181333

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 13 2010

Status

approved