A167635 Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at odd level.

1, 0, 1, 0, 2, 0, 5, 1, 14, 7, 43, 36, 143, 166, 509, 731, 1915, 3158, 7523, 13560, 30537, 58257, 127029, 251266, 538253, 1089666, 2313121, 4754148, 10051130, 20868070, 44065633, 92132176, 194617333, 408971295, 864899013, 1824485600, 3864369141

Offset

0,5

Comments

a(n)=A167634(n,0).

Links

Table of n, a(n) for n=0..36.

Formula

G.f.: G = [1 + 2z - z^3 - sqrt(1 - 4z^2 - 2z^3 + z^6)]/[2z(1 + z - z^2)].

D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+5)*a(n-2) +(-2*n+7)*a(n-3) +3*a(n-4) +a(n-5) +(n-6)*a(n-6)=0. - R. J. Mathar, Jul 26 2022

Example

a(6)=5 because we have UUDDUUDDUUDD, UUDDUUUUDDDD, UUUUDDDDUUDD, UUUUDDUUDDDD, and UUUUUUDDDDDD.

Maple

G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z-z^2)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);

Crossrefs

Cf. A167634, A167638

Sequence in context: A243998 A290395 A082974 * A192426 A075603 A264357

Adjacent sequences: A167632 A167633 A167634 * A167636 A167637 A167638

Keyword

nonn

Author

Emeric Deutsch, Nov 08 2009

Status

approved