OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..839
Wikipedia, Counting lattice paths
MAPLE
b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
124*n^2-135*n+35)*b(n-1)+4*(n-1)^2*(4*n-5)*(4*n-3)*(15*n^2-4*n-12)*
b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
end:
a:= n-> ((binomial(n+n, n)/(n+1))^2-b(n))/2:
seq(a(n), n=0..21);
MATHEMATICA
A129123[n_] := Sum[(Binomial[n, k]-Binomial[n, k-1])^4, {k, 0, Floor[n/2]}];
a[n_] := (CatalanNumber[n]^2 - A129123[n])/2;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 07 2022
STATUS
approved