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A212387 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 7).

%I A212387 #26 Apr 15 2017 00:49:49
%S A212387 1,1,1,1,1,1,1,1,2,10,46,166,496,1288,3004,6437,12895,24583,45799,
%T A212387 87211,180235,420547,1087220,2941931,7927664,20705636,51886966,
%U A212387 124660576,288445186,648173927,1431655546,3156274456,7062245781,16256654077,38704049941,94853117381
%N A212387 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 7).
%C A212387 Lengths of descents are unrestricted.
%H A212387 Alois P. Heinz, <a href="/A212387/b212387.txt">Table of n, a(n) for n = 0..1000</a>
%H A212387 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212387 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^7).
%F A212387 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 = 7 and r = 0.4020785148135889828..., s = 1.877947072112206660... 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 A212387 a(n) = Sum_{k=0..n} (binomial(6*k-5*n-1,n-k)*binomial(n+1,7*k-6*n))/(n+1). - _Vladimir Kruchinin_, Mar 05 2016
%e A212387 a(0) = 1: the empty path.
%e A212387 a(1) = 1: UD.
%e A212387 a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
%e A212387 a(9) = 10: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDDDDDDDUDD, UUUUUUUUDDDDDDUDDD, UUUUUUUUDDDDDUDDDD, UUUUUUUUDDDDUDDDDD, UUUUUUUUDDDUDDDDDD, UUUUUUUUDDUDDDDDDD, UUUUUUUUDUDDDDDDDD.
%p A212387 b:= proc(x, y, u) option remember;
%p A212387       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212387       `if`(u, add (b(x-(7*t+1), y, false), t=0..(x-1)/7), 0)))
%p A212387     end:
%p A212387 a:= n-> b(n$2, true):
%p A212387 seq(a(n), n=0..40);
%p A212387 # second Maple program
%p A212387 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^7), A), x, n+1), x, n):
%p A212387 seq(a(n), n=0..40);
%t A212387 a[n_] := Sum[Binomial[6k-5n-1, n-k]*Binomial[n+1, 7k-6n], {k, 0, n}]/(n+1);
%t A212387 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 03 2017, after _Vladimir Kruchinin_ *)
%o A212387 (Maxima)
%o A212387 a(n):=sum(binomial(6*k-5*n-1, n-k)*binomial(n+1, 7*k-6*n), k, 0, n)/(n+1); /* _Vladimir Kruchinin_, Mar 05 2016 */
%Y A212387 Column k=7 of A212382.
%K A212387 nonn
%O A212387 0,9
%A A212387 _Alois P. Heinz_, May 12 2012