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
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Sep 27 2013
STATUS
approved