A372646 - OEIS

%I #17 May 14 2024 02:25:04

%S 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,

%T 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,

%U 8,20,8,0,6,2,0,2,3,0,14,22,5,0,1,0,0,6

%N 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).

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

%H John Tyler Rascoe, <a href="/A372646/b372646.txt">Rows n = 0..70, flattened</a>

%H John Tyler Rascoe, <a href="/A372646/a372646_1.py.txt">Python program</a>.

%F 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.

%e T(10,4) = 2: (1,2,3,4), (4,3,2,1).

%e T(10,5) = 2: (2,1,2,3,2), (2,3,2,1,2).

%e T(10,7) = 1: (1,2,1,2,1,2,1).

%e Triangle T(n,k) begins:

%e   0;

%e   .  1;

%e   .  0;

%e   .  .  2;

%e   .  .  0, 1;

%e   .  .  0, 1;

%e   .  .  .  2, 2;

%e   .  .  .  0, 0, 1;

%e   .  .  .  0, 4, 1;

%e   .  .  .  0, 0, 3, 2;

%e   .  .  .  .  2, 2, 0, 1;

%e   ...

%o (Python) # see linked program

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

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

%K nonn,tabf

%O 0,4

%A _John Tyler Rascoe_, May 08 2024