A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 5, 2, 1, 1, 9, 9, 5, 2, 1, 1, 12, 16, 10, 5, 2, 1, 1, 16, 25, 19, 10, 5, 2, 1, 1, 20, 39, 32, 20, 10, 5, 2, 1, 1, 25, 56, 54, 35, 20, 10, 5, 2, 1, 1, 30, 80, 84, 61, 36, 20, 10, 5, 2, 1, 1, 36, 109, 129, 99, 64, 36, 20, 10, 5, 2, 1

Offset

1,5

Comments

A unimodal composition is a composition such that for some j, m, 1 <= x(1) <= x(2) <= ... <= x(j) >= x(j+1) >= ... >= x(m) >= 1.

Row sums are A001523.

T(2*n+1,n+1) = A000712(n) for n>=0. - Alois P. Heinz, Oct 03 2013

References

E. M. Wright, Stacks, Quart. J. Math. Oxford 19 (1968) 313-320, table s(r).

Links

Alois P. Heinz, rows n = 1..141, flattened

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

Formula

O.g.f. for column k: x^k/prod(i=1..k-1, 1-x^i )^2.

Example

1;
1,  1;
1,  2,  1;
1,  4,  2,  1;
1,  6,  5,  2,  1;
1,  9,  9,  5,  2,  1;
1, 12, 16, 10,  5,  2,  1;
1, 16, 25, 19, 10,  5,  2, 1;
1, 20, 39, 32, 20, 10,  5, 2, 1;
1, 25, 56, 54, 35, 20, 10, 5, 2, 1;
T(5,3) = 5 because we have: 3+2 = 2+3 = 3+1+1 = 1+3+1 = 1+1+3.

Maple

b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),
       j=t..min(k, n))), add(b(n-j, j, 0), j=1..min(t, n))))
    end:
T:= (n, k)-> b(n, 1, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Oct 03 2013

Mathematica

Map[Select[#, #>0&]&, Drop[Transpose[Table[CoefficientList[Series[x^n/(1-x^n)/Product[1-x^i, {i, 1, n-1}]^2, {x, 0, nn}], x], {n, 1, nn}]], 1]]//Grid

Crossrefs

Cf. A229707.

Sequence in context: A179438 A211970 A089688 * A319421 A092479 A124022

Adjacent sequences: A229703 A229704 A229705 * A229707 A229708 A229709

Keyword

nonn,tabl

Author

Geoffrey Critzer, Sep 27 2013

Status

approved