A181330 -id:A181330 - cp's OEIS frontend

A181331 Number of 0's in the top rows of all 2-compositions of n.

0, 1, 5, 23, 99, 408, 1632, 6388, 24596, 93488, 351664, 1311536, 4856432, 17873408, 65436544, 238480960, 865665600, 3131196672, 11290210560, 40594476800, 145588087552, 520933746688, 1860059009024, 6628828632064, 23582036472832

Offset: 0

Emeric Deutsch, Oct 13 2010

nonn

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.

			a(2)=5 because in (0/2), (1/1), (2,0), (1,0/0,1), (0,1/1,0), (1,1/0,0), and (0,0/1,1) (the 2-compositions are written as (top row / bottom row)) we have 1+0+1+1+1+0+2=5 zeros.

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.
Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-4).

Cf. A181330, A181294.

g := z*(1-z)^3/(1-4*z+2*z^2)^2: gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

LinearRecurrence[{8, -20, 16, -4}, {0, 1, 5, 23, 99}, 25] (* Georg Fischer, Feb 01 2021 *)

a(n) = Sum_{k=0..n} A181330(n,k).
a(n) = (1/2)*A181294(n).
G.f.: x*(1 - x)^3 / (1 - 4*x + 2*x^2)^2.
a(n) = A181292(n)-2*A181292(n-1)+A181292(n-2). - R. J. Mathar, Jul 24 2022

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A181331 Number of 0's in the top rows of all 2-compositions of n.

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