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%I #47 Jan 04 2024 10:57:35
%S 1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,0,1,1,1,0,1,0,0,
%T 1,0,0,2,1,0,0,2,1,1,0,0,1,1,1,1,0,1,3,2,0,0,3,2,1,1,1,0,2,3,2,1,0,0,
%U 3,4,3,1,0,0,4,4,3,2,1,0,4,6,4,2,0,0,6,7
%N Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).
%C T(n,k) is the number of integer compositions of n with first part 1, last part k, and all adjacent differences in {-1,1}. - _John Tyler Rascoe_, Aug 14 2023
%H Seiichi Manyama, <a href="/A291904/b291904.txt">Rows n = 0..481, flattened</a>
%F From _John Tyler Rascoe_, Aug 14 2023: (Start)
%F This triangle is T_1(n,k) of the general triangle T_m(n,k) for compositions of this kind with first part m.
%F T_m(n,k) for 0 < m, 0 <= n, and 0 <= k <= A003056(n+A000217(m-1)).
%F T_m(0,0) = T_m(m,m) = 1.
%F T_m(n,k) = T_m(n-k,k-1) + T_m(n-k,k+1) for m < n and 0 < k <= A003056(n+A000217(m-1)).
%F T_m(n,k) = 0 for 0 < n < m or n < k.
%F T_m(n,0) = 0 for 0 < n. (End)
%e First few rows are:
%e 1;
%e 0, 1;
%e 0, 0;
%e 0, 0, 1;
%e 0, 1, 0;
%e 0, 0, 0;
%e 0, 0, 1, 1;
%e 0, 1, 0, 0;
%e 0, 0, 1, 0;
%e 0, 1, 1, 1;
%e 0, 1, 0, 0, 1;
%e 0, 0, 2, 1, 0;
%e 0, 2, 1, 1, 0.
%t T[0, 0] = 1; T[_, 0] = 0; T[n_?Positive, k_] /; 0 < k <= Floor[(Sqrt[8n+1] - 1)/2] := T[n, k] = T[n-k, k-1] + T[n-k, k+1]; T[_, _] = 0;
%t Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1] - 1)/2]}] // Flatten (* _Jean-François Alcover_, May 29 2019 *)
%Y Row sums give A291905.
%Y Columns 0-1 give A000007, A227310 (for n>0).
%Y Cf. A008289, A173258, A291895, A291929.
%K nonn,tabf
%O 0,38
%A _Seiichi Manyama_, Sep 05 2017
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