OFFSET
1,5
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
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
O.g.f. for column k: x^k * prod(i=1..k-1, 1 + x^i)^2.
EXAMPLE
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.
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+1..min(k, n))), add(b(n-j, j, 0), j=1..min(t-1, n))))
end:
T:= (n, k)-> 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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Sep 27 2013
STATUS
approved