A357871 - OEIS

%I #25 Nov 19 2022 03:36:59

%S 1,1,2,5,21,183,3424,155833,25962389,10152021001,18355563410823,

%T 94826525443572702,1720192707342762602561,135432808172830648285721490,

%U 25492564910167901918236137649748,28315683468644276652408152922412713937,65407605920313732627652296139090181364409413

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

%H Alois P. Heinz, <a href="/A357871/b357871.txt">Table of n, a(n) for n = 0..63</a>

%H Vaclav Kotesovec, <a href="/A357871/a357871.jpg">Graph - the asymptotic ratio (5000 terms)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiset">Multiset</a>

%F a(n) = Sum_{j=0..floor(n/2)} binomial(A008315(n,j)+n-1,n).

%F From _Vaclav Kotesovec_, Nov 19 2022: (Start)

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

%F 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)

%e a(3) = 5:

%e   {///, ///, ///},

%e   {//\, //\, //\},

%e   {//\, //\, /\/},

%e   {//\, /\/, /\/},

%e   {/\/, /\/, /\/}.

%p b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,

%p      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))

%p     end:

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

%p seq(a(n), n=0..16);

%t 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}}]]];

%t a[n_] := Sum[Binomial[b[n, n-2*j]+n-1, n], {j, 0, n/2}];

%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Nov 17 2022, after _Alois P. Heinz_ *)

%t 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 *)

%Y Cf. A008315, A357825.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 17 2022