A318945
Triangle read by rows: T(n,k) (n>=2, 0 <= k <= n-2) = number of Dyck paths with k valleys of altitude k.
Original entry on oeis.org
1, 4, 1, 13, 5, 1, 39, 19, 6, 1, 112, 64, 26, 7, 1, 313, 201, 97, 34, 8, 1, 859, 603, 331, 139, 43, 9, 1, 2328, 1752, 1064, 512, 191, 53, 10, 1
Offset: 2
Triangle begins:
1,
4,1,
13,5,1,
39,19,6,1,
112,64,26,7,1,
313,201,97,34,8,1,
859,603,331,139,43,9,1,
2328,1752,1064,512,191,53,10,1,
...
- Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L., Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Mathematics (2018), 341(10), 2789-2807.
A320905
T(n, k) = binomial(2*n - 1 - k, k - 1)*hypergeom([2, 2, 1-k], [1, 1 - 2*k + 2*n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.
Original entry on oeis.org
1, 1, 5, 1, 7, 18, 1, 9, 31, 56, 1, 11, 48, 111, 160, 1, 13, 69, 198, 351, 432, 1, 15, 94, 325, 699, 1023, 1120, 1, 17, 123, 500, 1280, 2223, 2815, 2816, 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912, 1, 21, 193, 1026, 3525, 8330, 14198, 18324, 18943, 16640
Offset: 1
Triangle starts:
[1] 1
[2] 1, 5
[3] 1, 7, 18
[4] 1, 9, 31, 56
[5] 1, 11, 48, 111, 160
[6] 1, 13, 69, 198, 351, 432
[7] 1, 15, 94, 325, 699, 1023, 1120
[8] 1, 17, 123, 500, 1280, 2223, 2815, 2816
[9] 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912
Row sums with shifted indices in
A318947.
-
T := (n, k) -> binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1):
seq(seq(simplify(T(n, k)), k=1..n), n=1..10);
-
T[n_, k_] := Sum[Binomial[2*n-k, 2*n-2*k+1+j]*Binomial[j+2, 2],{j, 0, 2*n-k}]; Flatten[Table[T[n, k], {n, 1, 10}, {k, 1, n}]] (* Detlef Meya, Dec 31 2023 *)
-
T(n, k) = {sum(j=0, 2*n-k, binomial(2*n-k, 2*n - 2*k + 1 + j) * binomial(j+2, 2))} \\ Andrew Howroyd, Dec 31 2023
-
from functools import cache
@cache
def T(n, k):
if k < 1 or n < 1: return 0
if k == 1: return 1
if k == n: return n * (n + 3) * 2**(n - 3)
return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
for n in range(1, 10): print([T(n, k) for k in range(1, n+1)])
# after Detlef Meya, Peter Luschny, Jan 01 2024
Original entry on oeis.org
0, 1, 7, 33, 130, 461, 1525, 4802, 14577, 43025, 124226, 352437, 985821, 2725858, 7466185, 20291193, 54791842, 147164525, 393517477, 1048395650, 2784568545, 7377137441, 19503081602, 51470797413, 135641216685, 357029910946, 938837513785, 2466747164937
Offset: 0
-
a[n_] := Sum[Sum[Binomial[2*n + 1 - k, 2*n + 2 - 2*k + j]*Binomial[j + 2, 2], {j, 0, 2*n + 1 - k}], {k, 0, n}]; Flatten[Table[a[n], {n, 0, 27}]] (* Detlef Meya, Jan 09 2024 *)
Showing 1-3 of 3 results.