A372646 Irregular triangle read by rows, T(n,k) is the number of integer compositions of n such that their set of adjacent differences is a subset of {-1,1}, they contain 1 as a part, and have k parts. T(n,k) for n >= 0, floor(sqrt(2*(n+1))-(1/2)) <= k <= floor((2*n+1)/3).

0, 1, 0, 2, 0, 1, 0, 1, 2, 2, 0, 0, 1, 0, 4, 1, 0, 0, 3, 2, 2, 2, 0, 1, 0, 3, 6, 1, 0, 2, 0, 4, 2, 0, 2, 8, 3, 0, 1, 0, 0, 0, 6, 8, 1, 2, 8, 5, 0, 5, 2, 0, 0, 7, 14, 4, 0, 1, 0, 4, 6, 0, 10, 10, 1, 0, 0, 8, 20, 8, 0, 6, 2, 0, 2, 3, 0, 14, 22, 5, 0, 1, 0, 0, 6

Offset: 0

Author

John Tyler Rascoe, May 08 2024

Keywords

• nonn
• tabf

Comments

Is there a bijection between the unrestricted compositions of k-1 and compositions of this kind with k parts for k > 0?

Examples

			T(10,4) = 2: (1,2,3,4), (4,3,2,1).
T(10,5) = 2: (2,1,2,3,2), (2,3,2,1,2).
T(10,7) = 1: (1,2,1,2,1,2,1).
Triangle T(n,k) begins:
  0;
  .  1;
  .  0;
  .  .  2;
  .  .  0, 1;
  .  .  0, 1;
  .  .  .  2, 2;
  .  .  .  0, 0, 1;
  .  .  .  0, 4, 1;
  .  .  .  0, 0, 3, 2;
  .  .  .  .  2, 2, 0, 1;
  ...
		

Links

• John Tyler Rascoe, Rows n = 0..70, flattened
• John Tyler Rascoe, Python program.

Crossrefs

Cf. A131577 (empirical column sums), A372647 (row sums).

Cf. A003242, A173258, A227310, A309938, A364039.

Programs

Python

# see linked program

Formula

G.f. for k-th column is C(x,k) - (x^k)*C(x,k) for k > 0 where C(x,k) is the g.f of the k-th column of A309938.