A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..121, flattened

Wikipedia, Counting lattice paths

Formula

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

Example

Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

Crossrefs

Columns k=0-5,7 give A008619, A001405, A000108, A003161, A129123, A361887, A361890.

Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.

Main diagonal gives A357825.

Cf. A008315, A120730.

Sequence in context: A144434 A322057 A323767 * A159936 A305749 A320748

Adjacent sequences: A357821 A357822 A357823 * A357825 A357826 A357827

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 14 2022

Status

approved