A054146 -id:A054146 - cp's OEIS frontend

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

Emeric Deutsch, Oct 13 2010

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).

			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;

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.

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

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

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

A181307 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 2, 6, 1, 18, 6, 54, 27, 1, 162, 108, 10, 486, 405, 64, 1, 1458, 1458, 334, 14, 4374, 5103, 1549, 117, 1, 13122, 17496, 6652, 760, 18, 39366, 59049, 27064, 4238, 186, 1, 118098, 196830, 105796, 21324, 1450, 22, 354294, 649539, 401041, 99646, 9480, 271

Offset: 0

Emeric Deutsch, Oct 13 2010

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.

Number of entries in row n is 1+floor(n/2).

			T(2,1) = 1 because we have (1/1) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
  1;
  2;
  6,1;
  18,6;
  54,27,1;
  162,108,10;

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.

Cf. A003480, A008776, A054146.

G := (1-z)^2/(1-4*z+3*z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1-z)^2/(1-4*z+3*z^2-t*z^2).
G.f. of column k: z^(2*k)/((1-3*z)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
Sum_{k>=0} T(n,k) = A003480(n).
T(n,0) = 2*3^(n-1) = A008776(n-1).
Sum_{k>=0} k*T(n,k) = A054146(n-1).

cp's OEIS Frontend

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.

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