A346157 Total number of left-to-right maxima in Dyck paths of semilength n.

0, 1, 2, 6, 19, 63, 216, 758, 2705, 9777, 35698, 131425, 487201, 1816651, 6807742, 25621878, 96796225, 366902949, 1394851446, 5316835073, 20314772302, 77786795230, 298435201100, 1147019162326, 4415737088310, 17025146600174, 65732992038182, 254118443847070, 983579262641569

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Aubrey Blecher and Arnold Knopfmacher, Left to right maxima in Dyck Paths, arXiv:2107.03102 [math.CO], 2021. See Theorem 4 p. 8.

Formula

a(n) = Sum_{r=1..n} (d(r+1)-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+1)-d(r)), r=1..n))(numtheory[tau]):
seq(a(n), n=0..28);  # Alois P. Heinz, Jul 08 2021

Mathematica

a[n_] := (Sum[(DivisorSigma[0, r + 1] - DivisorSigma[0, r])*(Binomial[2*n - 1, n - r] - Binomial[2*n - 1, n - r - 1]), {r, 1, n}]);
Table[a[i], {i, 0, 28}] (* Kebbaj Mohamed Reda, Jun 06 2022 *)

Prog

(PARI) a(n) = sum(r=1, n, (numdiv(r+1)-numdiv(r))*(binomial(2*n-1, n-r)-binomial(2*n-1, n-r-1)));

Crossrefs

Cf. A000005, A007318, A346158, A346194.

Sequence in context: A109262 A006724 A057409 * A141771 A001170 A001168

Adjacent sequences: A346154 A346155 A346156 * A346158 A346159 A346160

Keyword

nonn

Author

Michel Marcus, Jul 08 2021

Status

approved