A291883 Number T(n,k) of symmetrically unique Dyck paths of semilength n and height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 9, 11, 4, 1, 0, 1, 19, 31, 19, 5, 1, 0, 1, 35, 91, 69, 29, 6, 1, 0, 1, 71, 250, 252, 127, 41, 7, 1, 0, 1, 135, 690, 855, 540, 209, 55, 8, 1, 0, 1, 271, 1863, 2867, 2117, 1005, 319, 71, 9, 1, 0, 1, 527, 5017, 9339, 8063, 4411, 1705, 461, 89, 10, 1

Offset

0,9

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) = (A080936(n,k) + A132890(n,k))/2.

Sum_{k=1..n} k * T(n,k) = A291886(n).

Example

: T(4,2) = 5:       /\      /\        /\/\    /\  /\    /\/\/\
:              /\/\/  \  /\/  \/\  /\/    \  /  \/  \  /      \
:
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,   1;
  0, 1,   5,   3,   1;
  0, 1,   9,  11,   4,   1;
  0, 1,  19,  31,  19,   5,   1;
  0, 1,  35,  91,  69,  29,   6,  1;
  0, 1,  71, 250, 252, 127,  41,  7, 1;
  0, 1, 135, 690, 855, 540, 209, 55, 8, 1;

Maple

b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y<x-1,
      b(x-1, y+1, max(y+1, k)), 0)+`if`(y>0, b(x-1, y-1, k), 0))
    end:
g:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0,
      g(x-2, y-1, k), 0)+ g(x-2, y+1, max(y+1, k)))
    end:
T:= n-> (p-> seq(coeff(p, z, i)/2, i=0..n))(b(2*n, 0$2)+g(2*n, 0$2)):
seq(T(n), n=0..14);

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y < x - 1, b[x - 1, y + 1, Max[y + 1, k]], 0] + If[y > 0, b[x - 1, y - 1, k], 0]];
g[x_, y_, k_] := g[x, y, k] = If[x == 0, z^k, If[y > 0, g[x - 2, y - 1, k], 0] + g[x - 2, y + 1, Max[y + 1, k]]];
T[n_] := Function[p, Table[Coefficient[p, z, i]/2, {i, 0, n}]][b[2*n, 0, 0] + g[2*n, 0, 0]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

Prog

(Python)
from sympy.core.cache import cacheit
from sympy import Poly, Symbol, flatten
z=Symbol('z')
@cacheit
def b(x, y, k): return z**k if x==0 else (b(x - 1, y + 1, max(y + 1, k)) if y<x - 1 else 0) + (b(x - 1, y - 1, k) if y>0 else 0)
@cacheit
def g(x, y, k): return z**k if x==0 else (g(x - 2, y - 1, k) if y>0 else 0) + g(x - 2, y + 1, max(y + 1, k))
def T(n): return 1 if n==0 else [i//2 for i in Poly(b(2*n, 0, 0) + g(2*n, 0, 0)).all_coeffs()[::-1]]
print(flatten(map(T, range(15)))) # Indranil Ghosh, Sep 06 2017

Crossrefs

Columns k=0-10 give: A000007, A057427, A056326, A291887, A291888, A291889, A291890, A291891, A291892, A291893, A291894.

Main and first two lower diagonals give A000012, A001477, A028387(n-1) for n>0.

Row sums give A007123(n+1).

T(2n,n) give A291885.

Cf. A080936, A132890, A291886.

Sequence in context: A370773 A119331 A351641 * A361957 A239145 A327127

Adjacent sequences: A291880 A291881 A291882 * A291884 A291885 A291886

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 05 2017

Status

approved