A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1, 1, 1, 10, 25, 141, 421, 1451, 3445, 6259, 6457, 2188, 1

Offset

0,9

Comments

A(n,k) is the number of Motzkin paths of length n with up steps in k colors. - Alois P. Heinz, Apr 12 2026

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).

A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).

(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

Example

Square array begins:
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   2,   3,    4,    5,    6,     7,     8, ...
   1,   4,   7,   10,   13,   16,    19,    22, ...
   1,   9,  21,   37,   57,   81,   109,   141, ...
   1,  21,  61,  121,  201,  301,   421,   561, ...
   1,  51, 191,  451,  861, 1451,  2251,  3291, ...
   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...
   ...

Maple

b:= proc(x, y, k) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, k*b(x-1, y+1, k)+b(x-1, y, k)+b(x-1, y-1, k)))
    end:
A:= (n, k)-> b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 14 2026

Mathematica

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Crossrefs

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.

Main diagonal gives A307906.

Cf. A000108, A107267, A247495, A307855.

Sequence in context: A173072 A219272 A084097 * A293991 A288638 A382100

Adjacent sequences: A306681 A306682 A306683 * A306685 A306686 A306687

Keyword

nonn,tabl

Author

Seiichi Manyama, May 06 2019

Status

approved