A088662 Number of peaks at even level in all symmetric Dyck paths of semilength n+2.

1, 2, 7, 12, 34, 60, 155, 280, 686, 1260, 2982, 5544, 12804, 24024, 54483, 102960, 230230, 437580, 967538, 1847560, 4047836, 7759752, 16871582, 32449872, 70100044, 135207800, 290473900, 561632400, 1200823560, 2326762800

Offset

0,2

Links

Table of n, a(n) for n=0..29.

Formula

G.f.: (1-2z+4z^3)/[2z^2*(1-2z)sqrt(1-4z^2)]-1/(2z^2).

a(2n) = (2n)!(2n^2+4n+1)/[n!(n+1)!], a(2n+1) = 2(2n+1)!/(n!)^2.

a(2n+1) = 2*A002457(n).

a(n) = (n!/4)*((1+(-1)^n)*(n^2+4*n+2)/((n/2)!*(n/2+1)!)+4*(1-(-1)^n)/(n/2-1/2)!^2). - Wesley Ivan Hurt, Jun 23 2015

D-finite with recurrence -(n+2)*(35*n-41)*a(n) -4*(n+1)*(n-9)*a(n-1) +4*(35*n^2+31*n-41)*a(n-2) +8*(2*n+23)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 24 2022

Maple

A088662:=n->(n!/4)*((1+(-1)^n)*(n^2+4*n+2)/((n/2)!*(n/2+1)!)+4*(1-(-1)^n)/(n/2-1/2)!^2): seq(A088662(n), n=0..40); # Wesley Ivan Hurt, Jun 23 2015

Mathematica

Table[(n!/4) ((1 + (-1)^n) (n^2 + 4 n + 2)/((n/2)! (n/2 + 1)!) + 4 (1 - (-1)^n)/(n/2 - 1/2)!^2), {n, 0, 40}] (* Wesley Ivan Hurt, Jun 23 2015 *)

Crossrefs

Cf. A002457.

Sequence in context: A308706 A059053 A032025 * A073710 A326026 A092831

Adjacent sequences: A088659 A088660 A088661 * A088663 A088664 A088665

Keyword

nonn,easy

Author

Emeric Deutsch, Nov 21 2003

Status

approved