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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.
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%I #22 Oct 11 2013 08:44:51

%S 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,

%T 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,

%U 0,0,0,7,20,20,14,9,6,3,2,1

%N 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.

%C 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.

%C Row sums are A059618.

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

%C T(2*n+1,n+1) = A022567(n) for n>=0. - _Alois P. Heinz_, Oct 11 2013

%H Alois P. Heinz, <a href="/A229707/b229707.txt">Rows n = 1..141, flattened</a>

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

%e 1,

%e 0, 1,

%e 0, 2, 1,

%e 0, 1, 2, 1,

%e 0, 0, 3, 2, 1,

%e 0, 0, 4, 3, 2, 1,

%e 0, 0, 3, 6, 3, 2, 1,

%e 0, 0, 2, 7, 6, 3, 2, 1,

%e 0, 0, 1, 8, 9, 6, 3, 2, 1,

%e 0, 0, 0, 10, 12, 9, 6, 3, 2, 1

%e T(7,3) = 3 because we have: 1+2+3+1 = 1+3+2+1 = 2+3+2.

%p b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

%p `if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),

%p j=t+1..min(k, n))), add(b(n-j, j, 0), j=1..min(t-1, n))))

%p end:

%p T:= (n, k)-> b(n, 0, k):

%p seq(seq(T(n, k), k=1..n), n=1..16); # _Alois P. Heinz_, Oct 07 2013

%t 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

%Y Cf. A229706.

%K nonn,tabl

%O 1,5

%A _Geoffrey Critzer_, Sep 27 2013