A357825 Total number of n-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j = 0..floor(n/2).

1, 1, 2, 9, 98, 4150, 562692, 211106945, 404883552194, 1766902576146876, 40519034229909243476, 2708397617879598970178238, 658332084097982587522119612196, 735037057881394837614680080889845116, 2030001034486747324990010196845670569155080

Offset: 0

Author

Alois P. Heinz, Oct 14 2022

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60
• Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)
• Wikipedia, Counting lattice paths

Crossrefs

Main diagonal of A357824.

Cf. A000108, A000225, A008315, A120730, A357871.

Programs

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-> add(b(n, n-2*j)^n, j=0..n/2):
seq(a(n), n=0..15);

Mathematica

Table[Sum[(Binomial[n, k]*(n - 2*k + 1)/(n - k + 1))^n, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 17 2022 *)

Formula

a(n) = A357824(n,n).

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

a(n) = Sum_{j=0..n} A120730(n,j)^n.

a(n) mod 2 = 1 <=> n in { A000225 }.

From Vaclav Kotesovec, Nov 17 2022: (Start)

a(n)^(1/n) ~ exp(-1/2) * 2^(n + 3/2) / (sqrt(Pi)*n).

Limit_{n->oo} a(n) / (2^(n^2 + 3*n/2) / (n^n * exp(n/2) * Pi^(n/2))) does not exist, see also graph. (End)

Conjecture: the superconguence a(2*p-1) == 1 (mod p^3) holds for all primes p >= 5 (checked up to p = 101). - Peter Bala, Mar 20 2023