A346158 Total number of weak left-to-right maxima in Dyck paths of semilength n.

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

Table of n, a(n) for n=0..27.

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

Cf. A000005, A007318, A346157, A346195.

Sequence in context: A247172 A339843 A148939 * A077587 A001893 A238979

Adjacent sequences: A346155 A346156 A346157 * A346159 A346160 A346161

Keyword

nonn

Author

Michel Marcus, Jul 08 2021

Status

approved