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A368579 Triangle read by rows. T(n, k) is the number of compositions of n where the first part k is the largest part and the last part is not 1.

%I A368579 #18 Jan 05 2024 12:45:58
%S A368579 1,-1,1,0,0,1,0,0,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,2,2,1,0,1,0,0,3,3,2,1,
%T A368579 0,1,0,0,5,6,4,2,1,0,1,0,0,8,11,7,4,2,1,0,1,0,0,13,20,14,8,4,2,1,0,1,
%U A368579 0,0,21,37,27,15,8,4,2,1,0,1
%N A368579 Triangle read by rows. T(n, k) is the number of compositions of n where the first part k is the largest part and the last part is not 1.
%F A368579 T(n, k) = F(k+1, n+1-k) - F(k+1, n-k) where F(k, n) = Sum_{j=1..min(n, k)} F(k, n-j) if n > 1 and otherwise n. F(n, k) refers to the generalized Fibonacci numbers A092921.
%e A368579 Triangle T(n, k) starts:
%e A368579   [0] [ 1]
%e A368579   [1] [-1, 1]
%e A368579   [2] [ 0, 0, 1]
%e A368579   [3] [ 0, 0, 0,  1]
%e A368579   [4] [ 0, 0, 1,  0, 1]
%e A368579   [5] [ 0, 0, 1,  1, 0, 1]
%e A368579   [6] [ 0, 0, 2,  2, 1, 0, 1]
%e A368579   [7] [ 0, 0, 3,  3, 2, 1, 0, 1]
%e A368579   [8] [ 0, 0, 5,  6, 4, 2, 1, 0, 1]
%e A368579   [9] [ 0, 0, 8, 11, 7, 4, 2, 1, 0, 1]
%e A368579 For instance, row 6 lists the compositions below:
%e A368579   0  .
%e A368579   1  .
%e A368579   2 [2, 2, 2], [2, 1, 1, 2];
%e A368579   3 [3, 3], [3, 1, 2];
%e A368579   4 [4, 2];
%e A368579   5  .
%e A368579   6 [6].
%o A368579 (Python)
%o A368579 from functools import cache
%o A368579 @cache
%o A368579 def F(k, n):
%o A368579     return sum(F(k, n-j) for j in range(1, min(k, n))) if n > 1 else n
%o A368579 def Trow(n):
%o A368579     return list(F(k+1, n+1-k) - F(k+1, n-k) for k in range(n+1))
%o A368579 print(flatten([Trow(n) for n in range(12)]))
%Y A368579 Cf. A368279 (row sums), A092921 (generalized Fibonacci), A000045 (Fibonacci column k=2), A034008 (T(2n, n)).
%K A368579 sign,tabl
%O A368579 0,24
%A A368579 _Peter Luschny_, Jan 05 2024