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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, May 28 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A003480, A181366, A181367.

Sequence in context: A181307 A008855 A181299 * A221913 A280370 A280980

Adjacent sequences:  A181362 A181363 A181364 * A181366 A181367 A181368

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Oct 15 2010

STATUS

approved

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Last modified August 4 20:00 EDT 2020. Contains 336202 sequences. (Running on oeis4.)