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A357652
Number of pairs of Dyck paths of semilength n such that the midpoint of the first is not below the midpoint of the second.
3
1, 1, 3, 21, 147, 1323, 12618, 131085, 1430187, 16297347, 191987562, 2325379147, 28821761290, 364290802138, 4682375323044, 61067639131197, 806671205158587, 10776418254992139, 145413196382253114, 1979833455619072515, 27174458892459331530, 375722890152963114330
OFFSET
0,3
LINKS
FORMULA
a(n) = (A001246(n) + A129123(n))/2 = (A000108(n)^2 + A129123(n))/2.
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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 07 2022
STATUS
approved