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.

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, 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, 0, 0, 21, 37, 27, 15, 8, 4, 2, 1, 0, 1

Offset: 0

Author

Peter Luschny, Jan 05 2024

Keywords

• sign
• tabl

Examples

			Triangle T(n, k) starts:
  [0] [ 1]
  [1] [-1, 1]
  [2] [ 0, 0, 1]
  [3] [ 0, 0, 0,  1]
  [4] [ 0, 0, 1,  0, 1]
  [5] [ 0, 0, 1,  1, 0, 1]
  [6] [ 0, 0, 2,  2, 1, 0, 1]
  [7] [ 0, 0, 3,  3, 2, 1, 0, 1]
  [8] [ 0, 0, 5,  6, 4, 2, 1, 0, 1]
  [9] [ 0, 0, 8, 11, 7, 4, 2, 1, 0, 1]
For instance, row 6 lists the compositions below:
  0  .
  1  .
  2 [2, 2, 2], [2, 1, 1, 2];
  3 [3, 3], [3, 1, 2];
  4 [4, 2];
  5  .
  6 [6].
		

Crossrefs

Cf. A368279 (row sums), A092921 (generalized Fibonacci), A000045 (Fibonacci column k=2), A034008 (T(2n, n)).

Programs

Python

from functools import cache
@cache
def F(k, n):
    return sum(F(k, n-j) for j in range(1, min(k, n))) if n > 1 else n
def Trow(n):
    return list(F(k+1, n+1-k) - F(k+1, n-k) for k in range(n+1))
print(flatten([Trow(n) for n in range(12)]))

Formula

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.