cp's OEIS Frontend

A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).

%I A212385 #34 Mar 06 2016 22:03:21
%S A212385 1,1,1,1,1,1,2,8,29,85,211,464,943,1873,3914,9101,23298,61915,162283,
%T A212385 409888,996456,2360486,5555333,13244114,32357022,80958851,205389082,
%U A212385 522000262,1317987172,3297123652,8190326857,20302864970,50482613327,126318440989
%N A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).
%C A212385 Lengths of descents are unrestricted.
%C A212385 The radius of convergence of g.f. A(x) is r = 5*(1-2*s+s^2)/(s*(5*s-4)) = 0.3804593157188..., where s = A(r) is described below. - _Vaclav Kotesovec_, Mar 20 2014
%H A212385 Alois P. Heinz, <a href="/A212385/b212385.txt">Table of n, a(n) for n = 0..1000</a>
%H A212385 Vaclav Kotesovec, <a href="/A212385/a212385.txt">Recurrence (of order 10)</a>
%H A212385 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212385 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5).
%F A212385 Representation in terms of special values of generalized hypergeometric function of type 12F11: a(n) = hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -(1/6)*n, -(1/6)*n+5/6, -(1/6)*n+2/3, -(1/6)*n+1/2, -(1/6)*n+1/3, 1/6-(1/6)*n], [1/6, 1/3, 1/3, 1/2, 1/2, 2/3, 2/3, 5/6, 5/6, 1, 7/6], 7^7/6^6), n>=0. - _Karol A. Penson_, Jun 21 2013
%F A212385 a(n) ~ s^(n+3/2) * (5*s-4)^(n+2) / (2 * sqrt(Pi) * sqrt(3*s-2) * n^(3/2) * 5^(n+5/2) * (s-1)^(2*n+9/2)), where s = 1.87696911628429... is the root of the equation 2869 - 29970*s + 138225*s^2 - 373000*s^3 + 655625*s^4 - 787500*s^5 + 656250*s^6 - 375000*s^7 + 140625*s^8 - 31250*s^9 + 3125*s^10 = 0. - _Vaclav Kotesovec_, Mar 20 2014
%F A212385 a(n) = Sum_{k=0..n}(binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1). - _Vladimir Kruchinin_, Mar 05 2016
%e A212385 a(0) = 1: the empty path.
%e A212385 a(1) = 1: UD.
%e A212385 a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
%e A212385 a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.
%p A212385 b:= proc(x, y, u) option remember;
%p A212385       `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212385       `if`(u, add(b(x-(5*t+1), y, false), t=0..(x-1)/5), 0)))
%p A212385     end:
%p A212385 a:= n-> b(n$2, true):
%p A212385 seq(a(n), n=0..40);
%p A212385 # second Maple program
%p A212385 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^5), A), x, n+1), x, n):
%p A212385 seq(a(n), n=0..40);
%t A212385 b[x_, y_, u_] := b[x, y, u] = If[x<0 || y<x, 0, If[x == 0 && y == 0, 1, b[x, y-1, True] + If[u, Sum [b[x - (5*t+1), y, False], {t, 0, (x-1)/5}], 0]]]; a[n_] := b[n, n, True]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%o A212385 (Maxima)
%o A212385 a(n):=sum(binomial(4*k-3*n-1, n-k)*binomial(n+1, 5*k-4*n), k, 0, n)/(n+1); /* _Vladimir Kruchinin_, Mar 05 2016 */
%o A212385 (PARI) a(n) = sum(k=0, n, binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1); \\ _Michel Marcus_, Mar 05 2016
%Y A212385 Column k=5 of A212382.
%K A212385 nonn
%O A212385 0,7
%A A212385 _Alois P. Heinz_, May 12 2012