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A364039 Triangle read by rows: T(n,k) is the number of integer compositions of n with first part k and differences between neighboring parts in {-1,1}.

%I A364039 #37 Sep 15 2023 06:12:00
%S A364039 1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,2,1,0,1,0,2,1,1,0,0,1,0,1,1,1,1,0,
%T A364039 0,1,0,1,3,2,0,0,0,0,1,0,3,2,1,2,1,0,0,0,1,0,2,3,2,1,0,0,0,0,0,1,0,3,
%U A364039 4,3,1,1,1,0,0,0,0,1,0,4,4,4,2,1,0,0,0,0,0,0,1
%N A364039 Triangle read by rows: T(n,k) is the number of integer compositions of n with first part k and differences between neighboring parts in {-1,1}.
%H A364039 Alois P. Heinz, <a href="/A364039/b364039.txt">Rows n = 0..200, flattened</a>
%F A364039 T(n,n) = 1.
%F A364039 T(n,k) = T(n-k,k+1) + T(n-k,k-1) for 0 < k < n.
%F A364039 T(n,k) = 0 for n < k.
%F A364039 T(n,0) = 0 for 0 < n.
%e A364039 Triangle begins:
%e A364039   1;
%e A364039   0, 1;
%e A364039   0, 0, 1;
%e A364039   0, 1, 1, 1;
%e A364039   0, 1, 0, 0, 1;
%e A364039   0, 0, 2, 1, 0, 1;
%e A364039   0, 2, 1, 1, 0, 0, 1;
%e A364039   0, 1, 1, 1, 1, 0, 0, 1;
%e A364039   0, 1, 3, 2, 0, 0, 0, 0, 1;
%e A364039   0, 3, 2, 1, 2, 1, 0, 0, 0, 1;
%e A364039   0, 2, 3, 2, 1, 0, 0, 0, 0, 0, 1;
%e A364039   ...
%e A364039 For n = 6 there are a total of 5 compositions:
%e A364039   T(6,1) = 2: (123), (1212)
%e A364039   T(6,2) = 1: (2121)
%e A364039   T(6,3) = 1: (321)
%e A364039   T(6,6) = 1: (6)
%p A364039 T:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
%p A364039      `if`(n=i, 1, add(T(n-i, i+j), j=[-1, 1])))
%p A364039     end: T(0$2):=1:
%p A364039 seq(seq(T(n, k), k=0..n), n=0..14);  # _Alois P. Heinz_, Aug 08 2023
%o A364039 (Python)
%o A364039 def A364039_rowlist(row_max):
%o A364039     A = []
%o A364039     for n in range(0,row_max+1):
%o A364039         A.append([])
%o A364039         for k in range(0,n+1):
%o A364039             z = 0
%o A364039             if n==k: z += 1
%o A364039             elif k > 1 and k-1 <= n-k: z += A[n-k][k-1]
%o A364039             if k+1 <= n-k and k != 0: z += A[n-k][k+1]
%o A364039             A[n].append(z)
%o A364039         print(A[n])
%o A364039 A364039_rowlist(12)
%Y A364039 Cf. A291905 (column k=1), A173258 (row sums).
%Y A364039 Cf. A227310, A291904, A309938, A362500, A364529.
%K A364039 nonn,easy,tabl
%O A364039 0,18
%A A364039 _John Tyler Rascoe_, Aug 06 2023