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 A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9). 2
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 12, 67, 287, 1002, 3004, 8009, 19449, 43759, 92380, 184787, 353137, 650497, 1170632, 2110021, 3977161, 8271836, 19536661, 51111062, 140210129, 385123916, 1032218316, 2670065961, 6645249777, 15922990909, 36823807747, 82485177457 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS Lengths of descents are unrestricted. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..800 Vaclav Kotesovec, Asymptotic of subsequences of A212382 FORMULA G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9). a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 9 and r = 0.4164039515514120671..., s = 1.882616423435763466... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014 EXAMPLE a(0) = 1: the empty path. a(1) = 1: UD. a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD. a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD. MAPLE b:= proc(x, y, u) option remember;       `if`(x<0 or  y b(n\$2, true): seq(a(n), n=0..40); # second Maple program: a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n): seq(a(n), n=0..40); CROSSREFS Column k=9 of A212382. Sequence in context: A039633 A265451 A180195 * A241074 A020062 A185035 Adjacent sequences:  A212386 A212387 A212388 * A212390 A212391 A212392 KEYWORD nonn AUTHOR Alois P. Heinz, May 12 2012 STATUS approved

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Last modified April 24 20:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)