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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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
LINKS
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
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.
Sequence in context: A370773 A119331 A351641 * A361957 A239145 A327127
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 05 2017
STATUS
approved

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Last modified April 14 18:28 EDT 2024. Contains 371667 sequences. (Running on oeis4.)