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A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.

%I A229706 #27 Mar 07 2015 15:31:20
%S A229706 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,
%T A229706 19,10,5,2,1,1,20,39,32,20,10,5,2,1,1,25,56,54,35,20,10,5,2,1,1,30,80,
%U A229706 84,61,36,20,10,5,2,1,1,36,109,129,99,64,36,20,10,5,2,1
%N A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.
%C A229706 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.
%C A229706 Row sums are A001523.
%C A229706 T(2*n+1,n+1) = A000712(n) for n>=0. - _Alois P. Heinz_, Oct 03 2013
%D A229706 E. M. Wright, Stacks, Quart. J. Math. Oxford 19 (1968) 313-320, table s(r).
%H A229706 Alois P. Heinz, <a href="/A229706/b229706.txt">rows n = 1..141, flattened</a>
%H A229706 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 46
%F A229706 O.g.f. for column k: x^k/prod(i=1..k-1, 1-x^i )^2.
%e A229706 1;
%e A229706 1,  1;
%e A229706 1,  2,  1;
%e A229706 1,  4,  2,  1;
%e A229706 1,  6,  5,  2,  1;
%e A229706 1,  9,  9,  5,  2,  1;
%e A229706 1, 12, 16, 10,  5,  2,  1;
%e A229706 1, 16, 25, 19, 10,  5,  2, 1;
%e A229706 1, 20, 39, 32, 20, 10,  5, 2, 1;
%e A229706 1, 25, 56, 54, 35, 20, 10, 5, 2, 1;
%e A229706 T(5,3) = 5 because we have: 3+2 = 2+3 = 3+1+1 = 1+3+1 = 1+1+3.
%p A229706 b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A229706       `if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),
%p A229706        j=t..min(k, n))), add(b(n-j, j, 0), j=1..min(t, n))))
%p A229706     end:
%p A229706 T:= (n, k)-> b(n, 1, k):
%p A229706 seq(seq(T(n, k), k=1..n), n=1..16);  # _Alois P. Heinz_, Oct 03 2013
%t A229706 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
%Y A229706 Cf. A229707.
%K A229706 nonn,tabl
%O A229706 1,5
%A A229706 _Geoffrey Critzer_, Sep 27 2013