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
Examples
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;
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.
Programs
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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 *)
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).
Comments