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
Examples
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;
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.
Programs
-
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 *)
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).
Comments