OFFSET
1,5
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).
EXAMPLE
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)
PROG
(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)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Aug 23 2019
STATUS
approved