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%I #17 Mar 06 2016 22:18:22
%S 1,1,1,1,1,1,1,1,1,1,1,2,13,79,365,1366,4369,12377,31825,75583,167961,
%T 352718,705466,1352585,2501205,4495351,7956391,14221936,26802361,
%U 56058016,133316626,350785307,967683665,2677259721,7246005881,18977267621,47931495649
%N Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 10).
%C Lengths of descents are unrestricted.
%H Alois P. Heinz, <a href="/A212390/b212390.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^10).
%F 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 = 10 and r = 0.421937635689419083..., s = 1.885352542104400040... 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
%e a(0) = 1: the empty path.
%e a(1) = 1: UD.
%e a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
%e a(12) = 13: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDDDDDDDDDDUDD, UUUUUUUUUUUDDDDDDDDDUDDD, UUUUUUUUUUUDDDDDDDDUDDDD, UUUUUUUUUUUDDDDDDDUDDDDD, UUUUUUUUUUUDDDDDDUDDDDDD, UUUUUUUUUUUDDDDDUDDDDDDD, UUUUUUUUUUUDDDDUDDDDDDDD, UUUUUUUUUUUDDDUDDDDDDDDD, UUUUUUUUUUUDDUDDDDDDDDDD, UUUUUUUUUUUDUDDDDDDDDDDD.
%p b:= proc(x, y, u) option remember;
%p `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p `if`(u, add(b(x-(10*t+1), y, false), t=0..(x-1)/10), 0)))
%p end:
%p a:= n-> b(n$2, true):
%p seq(a(n), n=0..40);
%p # second Maple program:
%p a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^10), A), x, n+1), x, n):
%p seq(a(n), n=0..40);
%Y Column k=10 of A212382.
%K nonn
%O 0,12
%A _Alois P. Heinz_, May 12 2012
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