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A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5). 7
1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 943, 1873, 3914, 9101, 23298, 61915, 162283, 409888, 996456, 2360486, 5555333, 13244114, 32357022, 80958851, 205389082, 522000262, 1317987172, 3297123652, 8190326857, 20302864970, 50482613327, 126318440989 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Lengths of descents are unrestricted.
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
LINKS
Vaclav Kotesovec, Recurrence (of order 10)
FORMULA
G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5).
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
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
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
EXAMPLE
a(0) = 1: the empty path.
a(1) = 1: UD.
a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.
MAPLE
b:= proc(x, y, u) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
`if`(u, add(b(x-(5*t+1), y, false), t=0..(x-1)/5), 0)))
end:
a:= n-> 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)^5), A), x, n+1), x, n):
seq(a(n), n=0..40);
MATHEMATICA
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 *)
PROG
(Maxima)
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 */
(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
CROSSREFS
Column k=5 of A212382.
Sequence in context: A306847 A364523 A107025 * A333882 A365760 A365758
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 12 2012
STATUS
approved

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Last modified March 3 11:44 EST 2024. Contains 370511 sequences. (Running on oeis4.)