OFFSET
2,4
COMMENTS
T(n, k) is also equal to the number of cornerless symmetric Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.7 and Proposition 3.8 at pp. 16 - 17 in Cho et al.).
LINKS
Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.
FORMULA
T(n, k) = Sum_{i=1..min(k,floor(n/2))} binomial(floor((k-1)/2), floor((i-1)/2))*binomial(floor(k/2), floor(i/2))*binomial(floor(n/2)+k-i, k). (See proposition 3.8 in Cho et al.).
T(4, n) = T(5, n) = A001651(n+1).
EXAMPLE
The array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 5, 7, 8, 10, 11, 13, ...
2, 4, 5, 7, 8, 10, 11, 13, ...
3, 9, 15, 27, 37, 55, 69, 93, ...
3, 9, 15, 27, 37, 55, 69, 93, ...
4, 16, 34, 76, 124, 216, 309, 471, ...
4, 16, 34, 76, 124, 216, 309, 471, ...
5, 25, 65, 175, 335, 675, 1095, 1875, ...
5, 25, 65, 175, 335, 675, 1095, 1875, ...
...
MATHEMATICA
T[n_, k_]:=Sum[Binomial[Floor[(k-1)/2], Floor[(i-1)/2]]Binomial[Floor[k/2], Floor[i/2]]Binomial[Floor[n/2]+k-i, k], {i, Min[k, Floor[n/2]]}]; Flatten[Table[T[n-k+1, k], {n, 2, 13}, {k, 1, n-1}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Jun 15 2022
STATUS
approved