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

%I A357871 #25 Nov 19 2022 03:36:59
%S A357871 1,1,2,5,21,183,3424,155833,25962389,10152021001,18355563410823,
%T A357871 94826525443572702,1720192707342762602561,135432808172830648285721490,
%U A357871 25492564910167901918236137649748,28315683468644276652408152922412713937,65407605920313732627652296139090181364409413
%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).
%H A357871 Alois P. Heinz, <a href="/A357871/b357871.txt">Table of n, a(n) for n = 0..63</a>
%H A357871 Vaclav Kotesovec, <a href="/A357871/a357871.jpg">Graph - the asymptotic ratio (5000 terms)</a>
%H A357871 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A357871 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiset">Multiset</a>
%F A357871 a(n) = Sum_{j=0..floor(n/2)} binomial(A008315(n,j)+n-1,n).
%F A357871 From _Vaclav Kotesovec_, Nov 19 2022: (Start)
%F A357871 a(n)^(1/n) ~ exp(1/2) * 2^(n + 3/2) / (sqrt(Pi) * n^2).
%F A357871 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 A357871 a(3) = 5:
%e A357871   {///, ///, ///},
%e A357871   {//\, //\, //\},
%e A357871   {//\, //\, /\/},
%e A357871   {//\, /\/, /\/},
%e A357871   {/\/, /\/, /\/}.
%p A357871 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A357871      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A357871     end:
%p A357871 a:= n-> add(binomial(b(n, n-2*j)+n-1, n), j=0..n/2):
%p A357871 seq(a(n), n=0..16);
%t A357871 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 A357871 a[n_] := Sum[Binomial[b[n, n-2*j]+n-1, n], {j, 0, n/2}];
%t A357871 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Nov 17 2022, after _Alois P. Heinz_ *)
%t A357871 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 A357871 Cf. A008315, A357825.
%K A357871 nonn
%O A357871 0,3
%A A357871 _Alois P. Heinz_, Oct 17 2022