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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

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-Fran├žois 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: A244372 A119331 A351641 * A239145 A327127 A151824

Adjacent sequences:  A291880 A291881 A291882 * A291884 A291885 A291886

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 05 2017

STATUS

approved

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Last modified June 30 06:00 EDT 2022. Contains 354914 sequences. (Running on oeis4.)