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
Keywords
Examples
a(6)=5 because we have UUDDUUDDUUDD, UUDDUUUUDDDD, UUUUDDDDUUDD, UUUUDDUUDDDD, and UUUUUUDDDDDD.
Programs
-
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);
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
Comments