A322291 -id:A322291 - cp's OEIS frontend

A375659 For 0<=k<=n, T(n,k) = the number of Dyck-type lattice paths of length n, starting at the point (0,k), triangle T read by rows.

1, 1, 2, 2, 3, 4, 3, 6, 7, 8, 6, 10, 14, 15, 16, 10, 20, 25, 30, 31, 32, 20, 35, 50, 56, 62, 63, 64, 35, 70, 91, 112, 119, 126, 127, 128, 70, 126, 182, 210, 238, 246, 254, 255, 256, 126, 252, 336, 420, 456, 492, 501, 510, 511, 512, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 1024

Offset: 0

Marilena Jianu, Aug 23 2024

A Dyck type lattice path has the steps (1,1) or (1,-1) and never passes below the x-axis.
For k>=n, the number of Dyck-type lattice paths is 2^n.
The sequence completes A322291 by adding a diagonal of powers of 2.

			  n | k=0    1    2    3    4    5    6    7
 ---+---------------------------------------
  0 |  1
  1 |  1    2
  2 |  2    3    4
  3 |  3    6    7    8
  4 |  6   10   14   15   16
  5 | 10   20   25   30   31   32
  6 | 20   35   50   56   62   63   64
  7 | 35   70   92  112  119  126  127  128

John Tyler Rascoe, Rows n = 0..140, flattened

T(n,0) = T(n-1,1) = A001405(n).
T(n,n) = A000079(n).
T(n,n-1) = A000225(n).
T(n,n-2) = A000918(n).
T(n,n-3) = A000247(n).
T(n,n-4) = A052515(n).
Row sums = A189162(n+1).
Cf. A322291, A368175.

a:=(n,k)->sum(binomial(n, floor((1/2)*(n-k))+i), i = 0..k):
seq(seq(a(n, k), k = 0..n), n = 0..11);

from math import comb
def A375659(n,k):
    return sum(comb(n,i+(n-k)//2) for i in range(k+1)) # John Tyler Rascoe, Sep 04 2024

T(n,k) = Sum_{i = 0..k} binomial(n, floor((n-k)/2)+i).

T(n,k) = T(n-1,k-1)+T(n-1,k+1), for all n>=2 and 1<=k<=n-2.

cp's OEIS Frontend

A375659 For 0<=k<=n, T(n,k) = the number of Dyck-type lattice paths of length n, starting at the point (0,k), triangle T read by rows.

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