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A346158
Total number of weak left-to-right maxima in Dyck paths of semilength n.
2
0, 1, 3, 9, 29, 98, 341, 1210, 4356, 15860, 58276, 215748, 803779, 3010514, 11327360, 42789497, 162200621, 616735888, 2351425517, 8987222524, 34425268864, 132128622038, 508050009119, 1956763431726, 7548022312030, 29156726321546, 112773951773678, 436719024636650
OFFSET
0,3
LINKS
Aubrey Blecher and Arnold Knopfmacher, Left to right maxima in Dyck Paths, arXiv:2107.03102 [math.CO], 2021. See Theorem 9 p. 12.
FORMULA
a(n) = Sum_{r=1..n} (d(r+2)-d(r))*(binomial(2*n-1,n-r)-binomial(2*n-1, n-r-1)) where d is A000005 and binomial is A007318.
MAPLE
a:= n-> (d-> add((binomial(2*n-1, n-r)-binomial(2*n-1, n-r-1))
*(d(r+2)-d(r)), r=1..n))(numtheory[tau]):
seq(a(n), n=0..27); # Alois P. Heinz, Jul 08 2021
PROG
(PARI) a(n) = sum(r=1, n, (numdiv(r+2)-numdiv(r))*(binomial(2*n-1, n-r)-binomial(2*n-1, n-r-1)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Jul 08 2021
STATUS
approved