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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.
2
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
1,5
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
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
Cf. A229706.
Sequence in context: A114114 A090787 A351559 * A262680 A366128 A191329
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Sep 27 2013
STATUS
approved