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A089198 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of non-squashing partitions of n into distinct parts of which the greatest is k.

%I A089198 #9 Feb 13 2019 07:53:12
%S A089198 1,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,2,1,
%T A089198 1,1,0,0,0,0,1,2,1,1,1,0,0,0,0,0,2,2,1,1,1,0,0,0,0,0,2,2,2,1,1,1,0,0,
%U A089198 0,0,0,0,3,2,2,1,1,1,0,0,0,0,0,0,3,3,2,2,1,1,1
%N A089198 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of non-squashing partitions of n into distinct parts of which the greatest is k.
%H A089198 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F A089198 The nonzero values of T(n, m) lie within a certain cone: T(n, m) = 0 if m < n/2 or if m > n. For m <= n <= 2m, T(n, m) = sum_{i=0}^{m-1} T(n-m, i).
%F A089198 For m <= n <= 2m, T(n, m) = b(n-m) if n < 2m, = b(n-m) - 1 if n = 2m, where b = A088567.
%e A089198 Triangle begins:
%e A089198 1
%e A089198 0 1
%e A089198 0 0 1
%e A089198 0 0 1 1
%e A089198 0 0 0 1 1
%e A089198 0 0 0 1 1 1
%e A089198 0 0 0 1 1 1 1
%e A089198 0 0 0 0 2 1 1 1
%e A089198 0 0 0 0 1 2 1 1 1
%t A089198 T[n_, m_] := T[n, m] = Which[n==m, 1, m<n/2 || m>n, 0, True, Sum[T[n-m, i], {i, 0, m-1}]];
%t A089198 Table[T[n, m], {n, 0, 12}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 13 2019 *)
%Y A089198 Row sums = A088567. Rows read from right to left also give (essentially) A088567.
%K A089198 nonn,tabl
%O A089198 0,33
%A A089198 _N. J. A. Sloane_, Dec 10 2003