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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 (list; table; graph; refs; listen; history; text; internal format)
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

Cf. A229706.

Sequence in context: A170968 A114114 A090787 * A262680 A191329 A096661

Adjacent sequences:  A229704 A229705 A229706 * A229708 A229709 A229710

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Sep 27 2013

STATUS

approved

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Last modified May 11 21:35 EDT 2021. Contains 343808 sequences. (Running on oeis4.)