A181327 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an even sum (0<=k<=floor(n/2)).

1, 2, 4, 3, 12, 12, 32, 41, 9, 86, 140, 54, 232, 451, 246, 27, 624, 1416, 1008, 216, 1680, 4357, 3811, 1215, 81, 4522, 13192, 13692, 5832, 810, 12172, 39455, 47380, 25254, 5400, 243, 32764, 116820, 159296, 102024, 29700, 2916, 88192, 343029, 523549

Offset

0,2

Comments

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.

Row n has 1+floor(n/2) entries.

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

T(n,0) = A181329(n).

Sum(k*T(n,k), k>=0) = A181328(n).

For the statistic "number of column with an odd sum" see A181308.

Links

Alois P. Heinz, Rows n = 0..200, 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/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4).

The g.f. of column k is z^{2k}*(1-z^2)^2*(3-z^2)^k/(1-2z-2z^2+z^4)^{k+1}

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(2,1) = 3 because we have (0 / 2), (1 / 1), and (2 / 0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
2;
4,    3;
12,  12;
32,  41,  9;
86, 140, 54;

Maple

G := (1-z^2)^2/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4): 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
# 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)=0, 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] == 0, 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, May 27 2015, after Alois P. Heinz *)

Crossrefs

Cf. A003480, A181308, A181328, A181329.

Sequence in context: A114883 A125091 A209048 * A257503 A091861 A200715

Adjacent sequences: A181324 A181325 A181326 * A181328 A181329 A181330

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 13 2010

Status

approved