cp's OEIS Frontend

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).

%I A372646 #17 May 14 2024 02:25:04
%S A372646 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 A372646 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 A372646 8,20,8,0,6,2,0,2,3,0,14,22,5,0,1,0,0,6
%N 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).
%C A372646 Is there a bijection between the unrestricted compositions of k-1 and compositions of this kind with k parts for k > 0?
%H A372646 John Tyler Rascoe, <a href="/A372646/b372646.txt">Rows n = 0..70, flattened</a>
%H A372646 John Tyler Rascoe, <a href="/A372646/a372646_1.py.txt">Python program</a>.
%F A372646 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 A372646 T(10,4) = 2: (1,2,3,4), (4,3,2,1).
%e A372646 T(10,5) = 2: (2,1,2,3,2), (2,3,2,1,2).
%e A372646 T(10,7) = 1: (1,2,1,2,1,2,1).
%e A372646 Triangle T(n,k) begins:
%e A372646   0;
%e A372646   .  1;
%e A372646   .  0;
%e A372646   .  .  2;
%e A372646   .  .  0, 1;
%e A372646   .  .  0, 1;
%e A372646   .  .  .  2, 2;
%e A372646   .  .  .  0, 0, 1;
%e A372646   .  .  .  0, 4, 1;
%e A372646   .  .  .  0, 0, 3, 2;
%e A372646   .  .  .  .  2, 2, 0, 1;
%e A372646   ...
%o A372646 (Python) # see linked program
%Y A372646 Cf. A131577 (empirical column sums), A372647 (row sums).
%Y A372646 Cf. A003242, A173258, A227310, A309938, A364039.
%K A372646 nonn,tabf
%O A372646 0,4
%A A372646 _John Tyler Rascoe_, May 08 2024