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A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).

%I A212386 #26 Apr 15 2017 00:46:41
%S A212386 1,1,1,1,1,1,1,2,9,37,121,331,793,1718,3454,6646,12841,26589,61813,
%T A212386 158918,426401,1134431,2914055,7171539,16967745,39008002,88529366,
%U A212386 202057561,471422866,1133448790,2799775102,7026467132,17684574313,44192085565,109081884957
%N A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).
%C A212386 Lengths of descents are unrestricted.
%H A212386 Alois P. Heinz, <a href="/A212386/b212386.txt">Table of n, a(n) for n = 0..1000</a>
%H A212386 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212386 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^6).
%F A212386 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 = 6 and r = 0.3925132712580446244..., s = 1.876653786643058101... 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
%F A212386 a(n) = Sum_{k=0..n} (binomial(5*k-4*n-1,n-k)*binomial(n+1,6*k-5*n))/(n+1). - _Vladimir Kruchinin_, Mar 05 2016
%e A212386 a(0) = 1: the empty path.
%e A212386 a(1) = 1: UD.
%e A212386 a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
%e A212386 a(8) = 9: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDDDDDDUDD, UUUUUUUDDDDDUDDD, UUUUUUUDDDDUDDDD, UUUUUUUDDDUDDDDD, UUUUUUUDDUDDDDDD, UUUUUUUDUDDDDDDD.
%p A212386 b:= proc(x, y, u) option remember;
%p A212386       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212386       `if`(u, add (b(x-(6*t+1), y, false), t=0..(x-1)/6), 0)))
%p A212386     end:
%p A212386 a:= n-> b(n$2, true):
%p A212386 seq(a(n), n=0..40);
%p A212386 # second Maple program
%p A212386 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^6), A), x, n+1), x, n):
%p A212386 seq(a(n), n=0..40);
%t A212386 a[n_] := Sum[Binomial[5k-4n-1, n-k]*Binomial[n+1, 6k-5n], {k, 0, n}]/(n+1);
%t A212386 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 03 2017, after _Vladimir Kruchinin_ *)
%o A212386 (Maxima)
%o A212386 a(n):=sum(binomial(5*k-4*n-1, n-k)*binomial(n+1, 6*k-5*n), k, 0, n)/(n+1); /* _Vladimir Kruchinin_, Mar 05 2016 */
%Y A212386 Column k=6 of A212382.
%K A212386 nonn
%O A212386 0,8
%A A212386 _Alois P. Heinz_, May 12 2012