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A212387
Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 7).
2
1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6437, 12895, 24583, 45799, 87211, 180235, 420547, 1087220, 2941931, 7927664, 20705636, 51886966, 124660576, 288445186, 648173927, 1431655546, 3156274456, 7062245781, 16256654077, 38704049941, 94853117381
OFFSET
0,9
COMMENTS
Lengths of descents are unrestricted.
FORMULA
G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^7).
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
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
EXAMPLE
a(0) = 1: the empty path.
a(1) = 1: UD.
a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
a(9) = 10: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDDDDDDDUDD, UUUUUUUUDDDDDDUDDD, UUUUUUUUDDDDDUDDDD, UUUUUUUUDDDDUDDDDD, UUUUUUUUDDDUDDDDDD, UUUUUUUUDDUDDDDDDD, UUUUUUUUDUDDDDDDDD.
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-(7*t+1), y, false), t=0..(x-1)/7), 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)^7), A), x, n+1), x, n):
seq(a(n), n=0..40);
MATHEMATICA
a[n_] := Sum[Binomial[6k-5n-1, n-k]*Binomial[n+1, 7k-6n], {k, 0, n}]/(n+1);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)
PROG
(Maxima)
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 */
CROSSREFS
Column k=7 of A212382.
Sequence in context: A306752 A306859 A373913 * A191813 A360412 A181294
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 12 2012
STATUS
approved