A181308 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an odd sum (0<=k<=n). 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, 0, 2, 3, 0, 4, 0, 16, 0, 8, 14, 0, 52, 0, 16, 0, 104, 0, 144, 0, 32, 64, 0, 460, 0, 368, 0, 64, 0, 616, 0, 1624, 0, 896, 0, 128, 292, 0, 3428, 0, 5056, 0, 2112, 0, 256, 0, 3456, 0, 14688, 0, 14528, 0, 4864, 0, 512, 1332, 0, 23132, 0, 53920, 0, 39488, 0, 11008, 0, 1024, 0

Offset

0,3

Comments

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

T(n,k) = 0 if n and k have opposite parities.

T(2n,0) = A060801(n).

Sum(k*T(n,k), k=0..n) = A181326(n).

For the statistic "number of column with an even sum" see A181327.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

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,z) = (1-z)^2*(1+z)^2/(1-5z^2+2z^4-2tz).

The g.f. of column k is (2z)^k*(1-z^2)^2/(1-5z^2+2z^4)^{k+1} (we have a Riordan array).

The g.f. H(t,s,z), where z marks size and t (s) marks number of columns with an odd (even) sum, is H=(1-z^2)^2/(1-2z^2+z^4-2tz-3sz^2+sz^4).

Example

T(2,2) = 4 because we have (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)).
Triangle starts:
1;
0,  2;
3,  0,  4;
0, 16,  0, 8;
14, 0, 52, 0, 16;

Maple

G := (1-z^2)^2/(1-5*z^2+2*z^4-2*t*z): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
       expand(add(add(`if`(i=0 and j=0, 0, b(n-i-j)*
       `if`(irem(i+j, 2)=1, x, 1)), i=0..n-j), j=0..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15); # Alois P. Heinz, Mar 16 2014

Mathematica

b[n_] := b[n] = If[n == 0, 1, Expand[Sum[Sum[If[i == 0 && j == 0, 0, b[n-i-j]* If[Mod[i+j, 2] == 1, x, 1]], {i, 0, n-j}], {j, 0, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Crossrefs

Cf. A003480, A060801, A181326, A181327.

Sequence in context: A293275 A014197 A341825 * A292246 A277141 A021438

Adjacent sequences: A181305 A181306 A181307 * A181309 A181310 A181311

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 13 2010

Status

approved