A309939 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and 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, 3, 6, 5, 1, 1, 2, 3, 6, 10, 6, 1, 1, 1, 3, 7, 12, 15, 7, 1, 1, 2, 3, 6, 14, 22, 21, 8, 1, 1, 1, 3, 8, 15, 27, 37, 28, 9, 1, 1, 2, 3, 6, 16, 32, 50, 58, 36, 10, 1, 1, 1, 3, 7, 16, 35, 63, 88, 86, 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, 3, 6,  5,  1;
  1, 2, 3, 6, 10,  6,  1;
  1, 1, 3, 7, 12, 15,  7,   1;
  1, 2, 3, 6, 14, 22, 21,   8,   1;
  1, 1, 3, 8, 15, 27, 37,  28,   9,   1;
  1, 2, 3, 6, 16, 32, 50,  58,  36,  10,   1;
  1, 1, 3, 7, 16, 35, 63,  88,  86,  45,  11,   1;
  1, 2, 3, 6, 16, 38, 74, 118, 147, 122,  55,  12,  1;
  1, 1, 3, 8, 16, 37, 83, 148, 212, 234, 167,  66, 13,  1;
  1, 2, 3, 6, 17, 40, 88, 174, 282, 366, 357, 222, 78, 14, 1;
  ...
For n = 6 there are a total of 17 compositions:
  k = 1: (6)
  k = 2: (33)
  k = 3: (123), (222), (321)
  k = 4: (1122), (1212), (1221), (2112), (2121), (2211)
  k = 5: (11112), (11121), (11211), (12111), (21111)
  k = 6: (111111)
		

Crossrefs

Row sums are A034297.

Cf. A309931, A309937, A309938.

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), R=matid(n), m=0); while(R, m++; v[m]+=vecsum(R[n,]); 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 for n > 1.

T(n, 3) = 3 for n > 3.

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

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