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

%I A357825 #36 Mar 23 2023 03:33:33
%S A357825 1,1,2,9,98,4150,562692,211106945,404883552194,1766902576146876,
%T A357825 40519034229909243476,2708397617879598970178238,
%U A357825 658332084097982587522119612196,735037057881394837614680080889845116,2030001034486747324990010196845670569155080
%N A357825 Total number of n-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j = 0..floor(n/2).
%H A357825 Alois P. Heinz, <a href="/A357825/b357825.txt">Table of n, a(n) for n = 0..60</a>
%H A357825 Vaclav Kotesovec, <a href="/A357825/a357825.jpg">Graph - the asymptotic ratio (5000 terms)</a>
%H A357825 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A357825 a(n) = A357824(n,n).
%F A357825 a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^n.
%F A357825 a(n) = Sum_{j=0..n} A120730(n,j)^n.
%F A357825 a(n) mod 2 = 1 <=> n in { A000225 }.
%F A357825 From _Vaclav Kotesovec_, Nov 17 2022: (Start)
%F A357825 a(n)^(1/n) ~ exp(-1/2) * 2^(n + 3/2) / (sqrt(Pi)*n).
%F A357825 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)
%F A357825 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
%p A357825 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A357825      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A357825     end:
%p A357825 a:= n-> add(b(n, n-2*j)^n, j=0..n/2):
%p A357825 seq(a(n), n=0..15);
%t A357825 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 *)
%Y A357825 Main diagonal of A357824.
%Y A357825 Cf. A000108, A000225, A008315, A120730, A357871.
%K A357825 nonn,easy
%O A357825 0,3
%A A357825 _Alois P. Heinz_, Oct 14 2022