A357825 -id:A357825 - cp's OEIS frontend

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

Alois P. Heinz, Oct 14 2022

			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, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

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);

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 *)

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).

A357871 Total number of n-multisets of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 5, 21, 183, 3424, 155833, 25962389, 10152021001, 18355563410823, 94826525443572702, 1720192707342762602561, 135432808172830648285721490, 25492564910167901918236137649748, 28315683468644276652408152922412713937, 65407605920313732627652296139090181364409413

Offset: 0

Alois P. Heinz, Oct 17 2022

nonn

			a(3) = 5:
  {///, ///, ///},
  {//\, //\, //\},
  {//\, //\, /\/},
  {//\, /\/, /\/},
  {/\/, /\/, /\/}.

Alois P. Heinz, Table of n, a(n) for n = 0..63
Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)
Wikipedia, Counting lattice paths
Wikipedia, Multiset

Cf. A008315, A357825.

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-> add(binomial(b(n, n-2*j)+n-1, n), j=0..n/2):
seq(a(n), n=0..16);

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_] := Sum[Binomial[b[n, n-2*j]+n-1, n], {j, 0, n/2}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)
Table[Sum[Binomial[Binomial[n,k]*(n-2*k+1)/(n-k+1) + n - 1,n], {k,0,n/2}],{n,0,16}] (* Vaclav Kotesovec, Nov 17 2022 *)

a(n) = Sum_{j=0..floor(n/2)} binomial(A008315(n,j)+n-1,n).
From Vaclav Kotesovec, Nov 19 2022: (Start)
a(n)^(1/n) ~ exp(1/2) * 2^(n + 3/2) / (sqrt(Pi) * n^2).
Limit_{n->infinity} a(n) / (exp(n/2) * 2^(n^2 + 3*n/2) / (Pi^(n/2) * n^(2*n + 1/2))) does not exist, see also graph. (End)

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula