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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.

%I A229707 #22 Oct 11 2013 08:44:51
%S A229707 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 A229707 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 A229707 0,0,0,7,20,20,14,9,6,3,2,1
%N 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.
%C A229707 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 A229707 Row sums are A059618.
%C A229707 Sum of column k is A000302(k-1).
%C A229707 T(2*n+1,n+1) = A022567(n) for n>=0. - _Alois P. Heinz_, Oct 11 2013
%H A229707 Alois P. Heinz, <a href="/A229707/b229707.txt">Rows n = 1..141, flattened</a>
%F A229707 O.g.f. for column k: x^k * prod(i=1..k-1, 1 + x^i)^2.
%e A229707 1,
%e A229707 0, 1,
%e A229707 0, 2, 1,
%e A229707 0, 1, 2, 1,
%e A229707 0, 0, 3, 2, 1,
%e A229707 0, 0, 4, 3, 2, 1,
%e A229707 0, 0, 3, 6, 3, 2, 1,
%e A229707 0, 0, 2, 7, 6, 3, 2, 1,
%e A229707 0, 0, 1, 8, 9, 6, 3, 2, 1,
%e A229707 0, 0, 0, 10, 12, 9, 6, 3, 2, 1
%e A229707 T(7,3) = 3 because we have: 1+2+3+1 = 1+3+2+1 = 2+3+2.
%p A229707 b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A229707       `if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),
%p A229707        j=t+1..min(k, n))), add(b(n-j, j, 0), j=1..min(t-1, n))))
%p A229707     end:
%p A229707 T:= (n, k)-> b(n, 0, k):
%p A229707 seq(seq(T(n, k), k=1..n), n=1..16);  # _Alois P. Heinz_, Oct 07 2013
%t A229707 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 A229707 Cf. A229706.
%K A229707 nonn,tabl
%O A229707 1,5
%A A229707 _Geoffrey Critzer_, Sep 27 2013