A309931 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and circular differences all equal to 1, 0, or -1.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 3, 4, 1, 1, 1, 1, 6, 5, 1, 1, 2, 3, 4, 10, 6, 1, 1, 1, 3, 5, 10, 15, 7, 1, 1, 2, 1, 4, 10, 20, 21, 8, 1, 1, 1, 3, 6, 11, 21, 35, 28, 9, 1, 1, 2, 3, 4, 10, 24, 42, 56, 36, 10, 1, 1, 1, 1, 5, 10, 25, 49, 78, 84, 45, 11, 1

Offset: 1

Author

Andrew Howroyd, Aug 23 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 3, 1;
  1, 2, 3, 4,  1;
  1, 1, 1, 6,  5,  1;
  1, 2, 3, 4, 10,  6,  1;
  1, 1, 3, 5, 10, 15,  7,   1;
  1, 2, 1, 4, 10, 20, 21,   8,   1;
  1, 1, 3, 6, 11, 21, 35,  28,   9,   1;
  1, 2, 3, 4, 10, 24, 42,  56,  36,  10,   1;
  1, 1, 1, 5, 10, 25, 49,  78,  84,  45,  11,  1;
  1, 2, 3, 4, 10, 24, 56,  96, 135, 120,  55, 12,  1;
  1, 1, 3, 6, 10, 21, 57, 116, 180, 220, 165, 66, 13, 1;
  ...
For n = 6 there are a total of 15 compositions:
  k = 1: (6)
  k = 2: (33)
  k = 3: (222)
  k = 4: (1122), (1212), (1221), (2112), (2121), (2211)
  k = 5: (11112), (11121), (11211), (12111), (21111)
  k = 6: (111111)
		

Crossrefs

Row sums are A325591.

Cf. A309937, A309938, A309939.

Programs

PARI

step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + R[i-j, j] + if(j+1<=n, R[i-j, j+1])) )}
T(n)={my(v=vector(n)); for(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)<=1), m=0); while(R, m++; v[m]+=R[n, k]; R=step(R, n))); v}
for(n=1, 12, print(T(n)));

Formula

T(n, 1) = T(n, n) = 1.

T(n, 2) = (3 - (-1)^n)/2.

T(n, n - 1) = binomial(n-1, 1) = n - 1.

T(n, n - 2) = binomial(n-2, 2).