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A167635
Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at odd level.
2
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).
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 08 2009
STATUS
approved