A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero entries 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, 2, 4, 1, 4, 12, 7, 1, 8, 32, 31, 10, 1, 16, 80, 111, 59, 13, 1, 32, 192, 351, 268, 96, 16, 1, 64, 448, 1023, 1037, 530, 142, 19, 1, 128, 1024, 2815, 3598, 2435, 924, 197, 22, 1, 256, 2304, 7423, 11535, 9843, 4923, 1477, 261, 25, 1, 512, 5120, 18943, 34832

Offset

0,4

Comments

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

T(n,1) = A001787(n).

T(n,2) = A055580(n-2) (n>=2).

T(n,3) = A055586(n-3) (n>=3).

Sum(k*T(n,k), k>=0) = A054146(n).

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

T(n,k) = sum(2^j*binomial(k+j,k)*binomial(n-2-j,k-2), j=0..n-k).

G.f.: G(t,x) = (1-x)^2/(1-3*x+2*x^2-t*x).

The g.f. of column k is x^k/((1-2*x)^(k+1)*(1-x)^(k-1)) (we have a Riordan array).

T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=2, T(2,1)=4, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham, Nov 26 2013

Example

T(2,1)=4 because we have (1/1), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
4,12,7,1;
8,32,31,10,1;
16,80,111,59,13,1;

Maple

T := proc (n, k) options operator, arrow: sum(2^j*binomial(k+j, k)*binomial(n-j-2, k-2), j = 0 .. n-k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A003480, A001787, A055580, A055586, A181330, A181332.

Sequence in context: A244261 A361873 A085111 * A204021 A236471 A220328

Adjacent sequences: A181329 A181330 A181331 * A181333 A181334 A181335

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 13 2010

Status

approved