A181365 Triangle read by rows: T(n,k) is the number of 2-compositions of n having least entry equal to k (n >= 1; 0 <= k <= floor(n/2)). 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.

2, 6, 1, 22, 2, 78, 3, 1, 272, 6, 2, 940, 13, 2, 1, 3232, 28, 2, 2, 11080, 58, 3, 2, 1, 37920, 118, 6, 2, 2, 129648, 239, 12, 2, 2, 1, 443008, 484, 22, 2, 2, 2, 1513248, 979, 37, 3, 2, 2, 1, 5168000, 1976, 60, 6, 2, 2, 2, 17647552, 3980, 97, 12, 2, 2, 2, 1, 60258304, 8004

Offset

1,1

Comments

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

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

T(n,1) = A181367(n).

Sum_{k >= 0} k*T(n,k) = A181366.

Links

Alois P. Heinz, Rows n = 1..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. for 2-compositions with all entries >= k is h(k,z)=(1-z)^2/(1-2z+z^2-z^{2k}) if k>0 and h(0,z)=(1-z)^2/(1-4z+2z^2) if k=0.

G.f. for 2-compositions with least entry k is f(k,z)=h(k,z)-h(k+1,z) (these are the column g.f.'s).

G.f.: G(t,z) = f(0,z) + Sum_{k>=1} f(k,z)*t^k.

Example

T(4,1) = 3 because we have (1/3), (3/1), and (1,1/1,1) (the 2-compositions are written as (top row / bottom row).
Triangle starts:
    2;
    6,  1;
   22,  2;
   78,  3, 1;
  272,  6, 2;
  940, 13, 2, 1;

Maple

h := proc (k) if k = 0 then (1-z)^2/(1-4*z+2*z^2) else (1-z)^2/(1-2*z+z^2-z^(2*k)) end if end proc: f := proc (k) options operator, arrow: h(k)-h(k+1) end proc; G := f(0)+sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 20)): for n to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
     `if`(i=0 and j=0, 0, A(n-i-j, k)), i=k..n-j), j=k..n))
    end:
T:= (n, k)-> A(n, k) -A(n, k+1):
seq(seq(T(n, k), k=0..n/2), n=1..15); # Alois P. Heinz, Mar 16 2014

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[If[i == 0 && j == 0, 0, A[n-i-j, k]], {i, k, n-j}], {j, k, n}]]; T[n_, k_] := A[n, k] - A[n, k+1]; Table[Table[T[n, k], {k, 0, n/2}], {n, 1, 15}] // Flatten (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Crossrefs

Cf. A003480, A181366, A181367.

Sequence in context: A008855 A345043 A181299 * A221913 A280370 A280980

Adjacent sequences: A181362 A181363 A181364 * A181366 A181367 A181368

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 15 2010

Status

approved