A229707 Triangular array read by rows. T(n,k) is the number of strictly unimodal compositions of n with the greatest part equal to k; n>=1, 1<=k<=n.

1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 3, 2, 1, 0, 0, 4, 3, 2, 1, 0, 0, 3, 6, 3, 2, 1, 0, 0, 2, 7, 6, 3, 2, 1, 0, 0, 1, 8, 9, 6, 3, 2, 1, 0, 0, 0, 10, 12, 9, 6, 3, 2, 1, 0, 0, 0, 8, 16, 14, 9, 6, 3, 2, 1, 0, 0, 0, 7, 20, 20, 14, 9, 6, 3, 2, 1

Offset: 1

Author

Geoffrey Critzer, Sep 27 2013

Keywords

• nonn
• tabl

Comments

A strictly 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 A059618.

Sum of column k is A000302(k-1).

T(2*n+1,n+1) = A022567(n) for n>=0. - Alois P. Heinz, Oct 11 2013

Examples

			1,
0, 1,
0, 2, 1,
0, 1, 2, 1,
0, 0, 3, 2, 1,
0, 0, 4, 3, 2, 1,
0, 0, 3, 6, 3, 2, 1,
0, 0, 2, 7, 6, 3, 2, 1,
0, 0, 1, 8, 9, 6, 3, 2, 1,
0, 0, 0, 10, 12, 9, 6, 3, 2, 1
T(7,3) = 3 because we have: 1+2+3+1 = 1+3+2+1 = 2+3+2.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A229706.

Programs

Maple

b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(k>0, `if`(n b(n, 0, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Oct 07 2013

Mathematica

nn=10;Table[Take[Drop[Transpose[Map[PadRight[#,nn+1,0]&,Table[CoefficientList[Series[x^n Product[(1+x^i),{i,1,n-1}]^2,{x,0,nn}],x],{n,1,nn}]]],1][[n]],n],{n,1,nn}]//Grid

Formula

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