A371705 -id:A371705 - cp's OEIS frontend

A371727 Number of Dyck paths of semilength n such that neighboring peaks differ in height by exactly one and first and last peak are at height one.

1, 1, 0, 0, 1, 0, 0, 2, 2, 1, 5, 12, 16, 28, 65, 128, 237, 478, 990, 2006, 4086, 8469, 17644, 36826, 77305, 163195, 345798, 735302, 1569379, 3360821, 7218566, 15548176, 33578893, 72698472, 157755230, 343071238, 747603060, 1632264655, 3570221869, 7822430724

Offset: 0

Alois P. Heinz, Apr 05 2024

nonn

			a(4) = 1:       /\
             /\/  \/\
.                                      /\
a(7) = 2:       /\    /\            /\/  \/\
             /\/  \/\/  \/\      /\/        \/\      .

Alois P. Heinz, Table of n, a(n) for n = 0..2764
Alois P. Heinz, Animation of a(16)=237 paths
Wikipedia, Counting lattice paths

Cf. A000108, A371705, A371726, A371926.

b:= proc(x, y, h, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, `if`(h>1, 0, 1), `if`(t=1 and abs(y-h)<>1, 0,
         b(x-1, y-1, `if`(t=1, y, h), 0))+b(x-1, y+1, h, 1)))
    end:
a:= n-> b(2*n, 0$3):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(x,y) option remember; (t->
     `if`(x=t, 1, `if`(x `if`(n=0, 1, b(2*n-1, 1)):
seq(a(n), n=0..50);

A371726 Number of Dyck paths of semilength n such that neighboring peaks differ in height by at most one and first and last peak are at height one.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 17, 39, 93, 227, 567, 1447, 3760, 9917, 26492, 71567, 195256, 537397, 1490614, 4163508, 11702411, 33078992, 93986117, 268294604, 769171970, 2213834911, 6395017291, 18534987580, 53887644375, 157121564325, 459351495858, 1346290327899

Offset: 0

Alois P. Heinz, Apr 04 2024

nonn

			a(4) = 2:                     /\
             /\/\/\/\      /\/  \/\      .

Alois P. Heinz, Table of n, a(n) for n = 0..2031
Alois P. Heinz, Animation of a(9)=93 paths
Wikipedia, Counting lattice paths

Cf. A000108, A287776, A371705, A371727.

b:= proc(x, y, h, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, `if`(h>1, 0, 1), `if`(t=1 and abs(y-h)>1, 0,
         b(x-1, y-1, `if`(t=1, y, h), 0))+b(x-1, y+1, h, 1)))
    end:
a:= n-> b(2*n, 0$3):
seq(a(n), n=0..32);

cp's OEIS Frontend

A371727 Number of Dyck paths of semilength n such that neighboring peaks differ in height by exactly one and first and last peak are at height one.

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